Theorem iota5 4360

Description: A method for computing iota. (Contributed by NM, 17-Sep-2013.)

Hypothesis

Ref	Expression
iota5.1	⊢ ((φ ∧ A ∈ V) → (ψ ↔ x = 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.

Step	Hyp	Ref	Expression
1		iota5.1	. . 3 ⊢ ((φ ∧ A ∈ V) → (ψ ↔ x = A))
2	1	alrimiv 1631	. 2 ⊢ ((φ ∧ A ∈ V) → ∀x(ψ ↔ x = A))
3		eqeq2 2362	. . . . . . 7 ⊢ (y = A → (x = y ↔ x = A))
4	3	bibi2d 309	. . . . . 6 ⊢ (y = A → ((ψ ↔ x = y) ↔ (ψ ↔ x = A)))
5	4	albidv 1625	. . . . 5 ⊢ (y = A → (∀x(ψ ↔ x = y) ↔ ∀x(ψ ↔ x = A)))
6		eqeq2 2362	. . . . 5 ⊢ (y = A → ((℩xψ) = y ↔ (℩xψ) = A))
7	5, 6	imbi12d 311	. . . 4 ⊢ (y = A → ((∀x(ψ ↔ x = y) → (℩xψ) = y) ↔ (∀x(ψ ↔ x = A) → (℩xψ) = A)))
8		iotaval 4351	. . . 4 ⊢ (∀x(ψ ↔ x = y) → (℩xψ) = y)
9	7, 8	vtoclg 2915	. . 3 ⊢ (A ∈ V → (∀x(ψ ↔ x = A) → (℩xψ) = A))
10	9	adantl 452	. 2 ⊢ ((φ ∧ A ∈ V) → (∀x(ψ ↔ x = A) → (℩xψ) = A))
11	2, 10	mpd 14	1 ⊢ ((φ ∧ A ∈ V) → (℩xψ) = A)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ℩cio 4338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2479  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-uni 3893  df-iota 4340

This theorem is referenced by: (None)