Theorem iota2 4367

Description: The unique element such that φ. (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Hypothesis

Ref	Expression
iota2.1	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
iota2	⊢ ((A ∈ B ∧ ∃!xφ) → (ψ ↔ (℩xφ) = A))

Distinct variable groups:   x,A   ψ,x

Allowed substitution hints:   φ(x)   B(x)

Proof of Theorem iota2

Step	Hyp	Ref	Expression
1		elex 2867	. 2 ⊢ (A ∈ B → A ∈ V)
2		simpl 443	. . 3 ⊢ ((A ∈ V ∧ ∃!xφ) → A ∈ V)
3		simpr 447	. . 3 ⊢ ((A ∈ V ∧ ∃!xφ) → ∃!xφ)
4		iota2.1	. . . 4 ⊢ (x = A → (φ ↔ ψ))
5	4	adantl 452	. . 3 ⊢ (((A ∈ V ∧ ∃!xφ) ∧ x = A) → (φ ↔ ψ))
6		nfv 1619	. . . 4 ⊢ Ⅎx A ∈ V
7		nfeu1 2214	. . . 4 ⊢ Ⅎx∃!xφ
8	6, 7	nfan 1824	. . 3 ⊢ Ⅎx(A ∈ V ∧ ∃!xφ)
9		nfvd 1620	. . 3 ⊢ ((A ∈ V ∧ ∃!xφ) → Ⅎxψ)
10		nfcvd 2490	. . 3 ⊢ ((A ∈ V ∧ ∃!xφ) → ℲxA)
11	2, 3, 5, 8, 9, 10	iota2df 4365	. 2 ⊢ ((A ∈ V ∧ ∃!xφ) → (ψ ↔ (℩xφ) = A))
12	1, 11	sylan 457	1 ⊢ ((A ∈ B ∧ ∃!xφ) → (ψ ↔ (℩xφ) = A))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Vcvv 2859  ℩cio 4337

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-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  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:  reiota2  4368  tz6.12-1  5344