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Theorem riota5f 5435
Description: A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
riota5f.1 (φxB)
riota5f.2 (φB A)
riota5f.3 ((φ x A) → (ψx = B))
Assertion
Ref Expression
riota5f (φ → (x A ψ) = B)
Distinct variable groups:   x,A   φ,x
Allowed substitution hints:   ψ(x)   B(x)

Proof of Theorem riota5f
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 riota5f.3 . . 3 ((φ x A) → (ψx = B))
21ralrimiva 2386 . 2 (φx A (ψx = B))
3 riota5f.2 . . . 4 (φB A)
4 a1tru 1258 . . . . . . 7 ((φ (y A x A (ψx = y))) → ⊤ )
5 reu6i 2726 . . . . . . . . 9 ((y A x A (ψx = y)) → ∃!x A ψ)
65adantl 262 . . . . . . . 8 ((φ (y A x A (ψx = y))) → ∃!x A ψ)
7 nfv 1418 . . . . . . . . . 10 xφ
8 nfv 1418 . . . . . . . . . . 11 x y A
9 nfra1 2349 . . . . . . . . . . 11 xx A (ψx = y)
108, 9nfan 1454 . . . . . . . . . 10 x(y A x A (ψx = y))
117, 10nfan 1454 . . . . . . . . 9 x(φ (y A x A (ψx = y)))
12 nfcvd 2176 . . . . . . . . 9 ((φ (y A x A (ψx = y))) → xy)
13 nfvd 1419 . . . . . . . . 9 ((φ (y A x A (ψx = y))) → Ⅎx ⊤ )
14 simprl 483 . . . . . . . . 9 ((φ (y A x A (ψx = y))) → y A)
15 simpr 103 . . . . . . . . . . 11 (((φ (y A x A (ψx = y))) x = y) → x = y)
16 simplrr 488 . . . . . . . . . . . 12 (((φ (y A x A (ψx = y))) x = y) → x A (ψx = y))
17 simplrl 487 . . . . . . . . . . . . 13 (((φ (y A x A (ψx = y))) x = y) → y A)
1815, 17eqeltrd 2111 . . . . . . . . . . . 12 (((φ (y A x A (ψx = y))) x = y) → x A)
19 rsp 2363 . . . . . . . . . . . 12 (x A (ψx = y) → (x A → (ψx = y)))
2016, 18, 19sylc 56 . . . . . . . . . . 11 (((φ (y A x A (ψx = y))) x = y) → (ψx = y))
2115, 20mpbird 156 . . . . . . . . . 10 (((φ (y A x A (ψx = y))) x = y) → ψ)
22 a1tru 1258 . . . . . . . . . 10 (((φ (y A x A (ψx = y))) x = y) → ⊤ )
2321, 222thd 164 . . . . . . . . 9 (((φ (y A x A (ψx = y))) x = y) → (ψ ↔ ⊤ ))
2411, 12, 13, 14, 23riota2df 5431 . . . . . . . 8 (((φ (y A x A (ψx = y))) ∃!x A ψ) → ( ⊤ ↔ (x A ψ) = y))
256, 24mpdan 398 . . . . . . 7 ((φ (y A x A (ψx = y))) → ( ⊤ ↔ (x A ψ) = y))
264, 25mpbid 135 . . . . . 6 ((φ (y A x A (ψx = y))) → (x A ψ) = y)
2726expr 357 . . . . 5 ((φ y A) → (x A (ψx = y) → (x A ψ) = y))
2827ralrimiva 2386 . . . 4 (φy A (x A (ψx = y) → (x A ψ) = y))
29 rspsbc 2834 . . . 4 (B A → (y A (x A (ψx = y) → (x A ψ) = y) → [B / y](x A (ψx = y) → (x A ψ) = y)))
303, 28, 29sylc 56 . . 3 (φ[B / y](x A (ψx = y) → (x A ψ) = y))
31 nfcvd 2176 . . . . . . . 8 (φxy)
32 riota5f.1 . . . . . . . 8 (φxB)
3331, 32nfeqd 2189 . . . . . . 7 (φ → Ⅎx y = B)
347, 33nfan1 1453 . . . . . 6 x(φ y = B)
35 simpr 103 . . . . . . . 8 ((φ y = B) → y = B)
3635eqeq2d 2048 . . . . . . 7 ((φ y = B) → (x = yx = B))
3736bibi2d 221 . . . . . 6 ((φ y = B) → ((ψx = y) ↔ (ψx = B)))
3834, 37ralbid 2318 . . . . 5 ((φ y = B) → (x A (ψx = y) ↔ x A (ψx = B)))
3935eqeq2d 2048 . . . . 5 ((φ y = B) → ((x A ψ) = y ↔ (x A ψ) = B))
4038, 39imbi12d 223 . . . 4 ((φ y = B) → ((x A (ψx = y) → (x A ψ) = y) ↔ (x A (ψx = B) → (x A ψ) = B)))
413, 40sbcied 2793 . . 3 (φ → ([B / y](x A (ψx = y) → (x A ψ) = y) ↔ (x A (ψx = B) → (x A ψ) = B)))
4230, 41mpbid 135 . 2 (φ → (x A (ψx = B) → (x A ψ) = B))
432, 42mpd 13 1 (φ → (x A ψ) = B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wtru 1243   wcel 1390  wnfc 2162  wral 2300  ∃!wreu 2302  [wsbc 2758  crio 5410
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-v 2553  df-sbc 2759  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810  df-riota 5411
This theorem is referenced by:  riota5  5436
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