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Theorem iota5 4359
 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 1631 . 2 ((φ A V) → x(ψx = A))
3 eqeq2 2362 . . . . . . 7 (y = A → (x = yx = A))
43bibi2d 309 . . . . . 6 (y = A → ((ψx = y) ↔ (ψx = A)))
54albidv 1625 . . . . 5 (y = A → (x(ψx = y) ↔ x(ψx = A)))
6 eqeq2 2362 . . . . 5 (y = A → ((℩xψ) = y ↔ (℩xψ) = A))
75, 6imbi12d 311 . . . 4 (y = A → ((x(ψx = y) → (℩xψ) = y) ↔ (x(ψx = A) → (℩xψ) = A)))
8 iotaval 4350 . . . 4 (x(ψx = y) → (℩xψ) = y)
97, 8vtoclg 2914 . . 3 (A V → (x(ψx = A) → (℩xψ) = A))
109adantl 452 . 2 ((φ A V) → (x(ψx = A) → (℩xψ) = A))
112, 10mpd 14 1 ((φ A V) → (℩xψ) = A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ℩cio 4337 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-uni 3892  df-iota 4339 This theorem is referenced by: (None)
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