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Theorem iota5 4814
 Description: A method for computing iota. (Contributed by NM, 17-Sep-2013.)
Hypothesis
Ref Expression
iota5.1 ((φ A 𝑉) → (ψx = A))
Assertion
Ref Expression
iota5 ((φ A 𝑉) → (℩xψ) = A)
Distinct variable groups:   x,A   x,𝑉   φ,x
Allowed substitution hint:   ψ(x)

Proof of Theorem iota5
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 iota5.1 . . 3 ((φ A 𝑉) → (ψx = A))
21alrimiv 1736 . 2 ((φ A 𝑉) → x(ψx = A))
3 eqeq2 2031 . . . . . . 7 (y = A → (x = yx = A))
43bibi2d 221 . . . . . 6 (y = A → ((ψx = y) ↔ (ψx = A)))
54albidv 1687 . . . . 5 (y = A → (x(ψx = y) ↔ x(ψx = A)))
6 eqeq2 2031 . . . . 5 (y = A → ((℩xψ) = y ↔ (℩xψ) = A))
75, 6imbi12d 223 . . . 4 (y = A → ((x(ψx = y) → (℩xψ) = y) ↔ (x(ψx = A) → (℩xψ) = A)))
8 iotaval 4805 . . . 4 (x(ψx = y) → (℩xψ) = y)
97, 8vtoclg 2590 . . 3 (A 𝑉 → (x(ψx = A) → (℩xψ) = A))
109adantl 262 . 2 ((φ A 𝑉) → (x(ψx = A) → (℩xψ) = A))
112, 10mpd 13 1 ((φ A 𝑉) → (℩xψ) = A)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1226   = wceq 1228   ∈ wcel 1374  ℩cio 4792 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-sn 3356  df-pr 3357  df-uni 3555  df-iota 4794 This theorem is referenced by: (None)
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