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Theorem reusv3 4158
Description: Two ways to express single-valuedness of a class expression 𝐶(y). See reusv1 4156 for the connection to uniqueness. (Contributed by NM, 27-Dec-2012.)
Hypotheses
Ref Expression
reusv3.1 (y = z → (φψ))
reusv3.2 (y = z𝐶 = 𝐷)
Assertion
Ref Expression
reusv3 (y B (φ 𝐶 A) → (y B z B ((φ ψ) → 𝐶 = 𝐷) ↔ x A y B (φx = 𝐶)))
Distinct variable groups:   x,y,z,B   x,𝐶,z   x,𝐷,y   φ,x,z   ψ,x,y   x,A,y,z
Allowed substitution hints:   φ(y)   ψ(z)   𝐶(y)   𝐷(z)

Proof of Theorem reusv3
StepHypRef Expression
1 reusv3.1 . . . . 5 (y = z → (φψ))
2 reusv3.2 . . . . . 6 (y = z𝐶 = 𝐷)
32eleq1d 2103 . . . . 5 (y = z → (𝐶 A𝐷 A))
41, 3anbi12d 442 . . . 4 (y = z → ((φ 𝐶 A) ↔ (ψ 𝐷 A)))
54cbvrexv 2528 . . 3 (y B (φ 𝐶 A) ↔ z B (ψ 𝐷 A))
6 nfra2xy 2358 . . . . 5 zy B z B ((φ ψ) → 𝐶 = 𝐷)
7 nfv 1418 . . . . 5 zx A y B (φx = 𝐶)
86, 7nfim 1461 . . . 4 z(y B z B ((φ ψ) → 𝐶 = 𝐷) → x A y B (φx = 𝐶))
9 risset 2346 . . . . . 6 (𝐷 Ax A x = 𝐷)
10 ralcom 2467 . . . . . . . . . . . . . 14 (y B z B ((φ ψ) → 𝐶 = 𝐷) ↔ z B y B ((φ ψ) → 𝐶 = 𝐷))
11 impexp 250 . . . . . . . . . . . . . . . . . 18 (((φ ψ) → 𝐶 = 𝐷) ↔ (φ → (ψ𝐶 = 𝐷)))
12 bi2.04 237 . . . . . . . . . . . . . . . . . 18 ((φ → (ψ𝐶 = 𝐷)) ↔ (ψ → (φ𝐶 = 𝐷)))
1311, 12bitri 173 . . . . . . . . . . . . . . . . 17 (((φ ψ) → 𝐶 = 𝐷) ↔ (ψ → (φ𝐶 = 𝐷)))
1413ralbii 2324 . . . . . . . . . . . . . . . 16 (y B ((φ ψ) → 𝐶 = 𝐷) ↔ y B (ψ → (φ𝐶 = 𝐷)))
15 r19.21v 2390 . . . . . . . . . . . . . . . 16 (y B (ψ → (φ𝐶 = 𝐷)) ↔ (ψy B (φ𝐶 = 𝐷)))
1614, 15bitri 173 . . . . . . . . . . . . . . 15 (y B ((φ ψ) → 𝐶 = 𝐷) ↔ (ψy B (φ𝐶 = 𝐷)))
1716ralbii 2324 . . . . . . . . . . . . . 14 (z B y B ((φ ψ) → 𝐶 = 𝐷) ↔ z B (ψy B (φ𝐶 = 𝐷)))
1810, 17bitri 173 . . . . . . . . . . . . 13 (y B z B ((φ ψ) → 𝐶 = 𝐷) ↔ z B (ψy B (φ𝐶 = 𝐷)))
19 rsp 2363 . . . . . . . . . . . . 13 (z B (ψy B (φ𝐶 = 𝐷)) → (z B → (ψy B (φ𝐶 = 𝐷))))
2018, 19sylbi 114 . . . . . . . . . . . 12 (y B z B ((φ ψ) → 𝐶 = 𝐷) → (z B → (ψy B (φ𝐶 = 𝐷))))
2120com3l 75 . . . . . . . . . . 11 (z B → (ψ → (y B z B ((φ ψ) → 𝐶 = 𝐷) → y B (φ𝐶 = 𝐷))))
2221imp31 243 . . . . . . . . . 10 (((z B ψ) y B z B ((φ ψ) → 𝐶 = 𝐷)) → y B (φ𝐶 = 𝐷))
23 eqeq1 2043 . . . . . . . . . . . . 13 (x = 𝐷 → (x = 𝐶𝐷 = 𝐶))
24 eqcom 2039 . . . . . . . . . . . . 13 (𝐷 = 𝐶𝐶 = 𝐷)
2523, 24syl6bb 185 . . . . . . . . . . . 12 (x = 𝐷 → (x = 𝐶𝐶 = 𝐷))
2625imbi2d 219 . . . . . . . . . . 11 (x = 𝐷 → ((φx = 𝐶) ↔ (φ𝐶 = 𝐷)))
2726ralbidv 2320 . . . . . . . . . 10 (x = 𝐷 → (y B (φx = 𝐶) ↔ y B (φ𝐶 = 𝐷)))
2822, 27syl5ibrcom 146 . . . . . . . . 9 (((z B ψ) y B z B ((φ ψ) → 𝐶 = 𝐷)) → (x = 𝐷y B (φx = 𝐶)))
2928reximdv 2414 . . . . . . . 8 (((z B ψ) y B z B ((φ ψ) → 𝐶 = 𝐷)) → (x A x = 𝐷x A y B (φx = 𝐶)))
3029ex 108 . . . . . . 7 ((z B ψ) → (y B z B ((φ ψ) → 𝐶 = 𝐷) → (x A x = 𝐷x A y B (φx = 𝐶))))
3130com23 72 . . . . . 6 ((z B ψ) → (x A x = 𝐷 → (y B z B ((φ ψ) → 𝐶 = 𝐷) → x A y B (φx = 𝐶))))
329, 31syl5bi 141 . . . . 5 ((z B ψ) → (𝐷 A → (y B z B ((φ ψ) → 𝐶 = 𝐷) → x A y B (φx = 𝐶))))
3332expimpd 345 . . . 4 (z B → ((ψ 𝐷 A) → (y B z B ((φ ψ) → 𝐶 = 𝐷) → x A y B (φx = 𝐶))))
348, 33rexlimi 2420 . . 3 (z B (ψ 𝐷 A) → (y B z B ((φ ψ) → 𝐶 = 𝐷) → x A y B (φx = 𝐶)))
355, 34sylbi 114 . 2 (y B (φ 𝐶 A) → (y B z B ((φ ψ) → 𝐶 = 𝐷) → x A y B (φx = 𝐶)))
361, 2reusv3i 4157 . 2 (x A y B (φx = 𝐶) → y B z B ((φ ψ) → 𝐶 = 𝐷))
3735, 36impbid1 130 1 (y B (φ 𝐶 A) → (y B z B ((φ ψ) → 𝐶 = 𝐷) ↔ x A y B (φx = 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  wral 2300  wrex 2301
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-tru 1245  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306
This theorem is referenced by: (None)
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