Theorem ceqsalt 2881

Description: Closed theorem version of ceqsalg 2883. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)

Assertion

Ref	Expression
ceqsalt	⊢ ((Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ)) ∧ A ∈ V) → (∀x(x = A → φ) ↔ ψ))

Distinct variable group:   x,A

Allowed substitution hints:   φ(x)   ψ(x)   V(x)

Proof of Theorem ceqsalt

Step	Hyp	Ref	Expression
1		elisset 2869	. . . 4 ⊢ (A ∈ V → ∃x x = A)
2	1	3ad2ant3 978	. . 3 ⊢ ((Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ)) ∧ A ∈ V) → ∃x x = A)
3		bi1 178	. . . . . . 7 ⊢ ((φ ↔ ψ) → (φ → ψ))
4	3	imim3i 55	. . . . . 6 ⊢ ((x = A → (φ ↔ ψ)) → ((x = A → φ) → (x = A → ψ)))
5	4	al2imi 1561	. . . . 5 ⊢ (∀x(x = A → (φ ↔ ψ)) → (∀x(x = A → φ) → ∀x(x = A → ψ)))
6	5	3ad2ant2 977	. . . 4 ⊢ ((Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ)) ∧ A ∈ V) → (∀x(x = A → φ) → ∀x(x = A → ψ)))
7		19.23t 1800	. . . . 5 ⊢ (Ⅎxψ → (∀x(x = A → ψ) ↔ (∃x x = A → ψ)))
8	7	3ad2ant1 976	. . . 4 ⊢ ((Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ)) ∧ A ∈ V) → (∀x(x = A → ψ) ↔ (∃x x = A → ψ)))
9	6, 8	sylibd 205	. . 3 ⊢ ((Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ)) ∧ A ∈ V) → (∀x(x = A → φ) → (∃x x = A → ψ)))
10	2, 9	mpid 37	. 2 ⊢ ((Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ)) ∧ A ∈ V) → (∀x(x = A → φ) → ψ))
11		bi2 189	. . . . . . 7 ⊢ ((φ ↔ ψ) → (ψ → φ))
12	11	imim2i 13	. . . . . 6 ⊢ ((x = A → (φ ↔ ψ)) → (x = A → (ψ → φ)))
13	12	com23 72	. . . . 5 ⊢ ((x = A → (φ ↔ ψ)) → (ψ → (x = A → φ)))
14	13	alimi 1559	. . . 4 ⊢ (∀x(x = A → (φ ↔ ψ)) → ∀x(ψ → (x = A → φ)))
15	14	3ad2ant2 977	. . 3 ⊢ ((Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ)) ∧ A ∈ V) → ∀x(ψ → (x = A → φ)))
16		19.21t 1795	. . . 4 ⊢ (Ⅎxψ → (∀x(ψ → (x = A → φ)) ↔ (ψ → ∀x(x = A → φ))))
17	16	3ad2ant1 976	. . 3 ⊢ ((Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ)) ∧ A ∈ V) → (∀x(ψ → (x = A → φ)) ↔ (ψ → ∀x(x = A → φ))))
18	15, 17	mpbid 201	. 2 ⊢ ((Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ)) ∧ A ∈ V) → (ψ → ∀x(x = A → φ)))
19	10, 18	impbid 183	1 ⊢ ((Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ)) ∧ A ∈ V) → (∀x(x = A → φ) ↔ ψ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ w3a 934  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710

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-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861

This theorem is referenced by:  ceqsralt  2882