Theorem ceqsalg 2884

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypotheses

Ref	Expression
ceqsalg.1	⊢ Ⅎxψ
ceqsalg.2	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
ceqsalg	⊢ (A ∈ V → (∀x(x = A → φ) ↔ ψ))

Distinct variable group:   x,A

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

Proof of Theorem ceqsalg

Step	Hyp	Ref	Expression
1		elisset 2870	. . 3 ⊢ (A ∈ V → ∃x x = A)
2		nfa1 1788	. . . 4 ⊢ Ⅎx∀x(x = A → φ)
3		ceqsalg.1	. . . 4 ⊢ Ⅎxψ
4		ceqsalg.2	. . . . . . 7 ⊢ (x = A → (φ ↔ ψ))
5	4	biimpd 198	. . . . . 6 ⊢ (x = A → (φ → ψ))
6	5	a2i 12	. . . . 5 ⊢ ((x = A → φ) → (x = A → ψ))
7	6	sps 1754	. . . 4 ⊢ (∀x(x = A → φ) → (x = A → ψ))
8	2, 3, 7	exlimd 1806	. . 3 ⊢ (∀x(x = A → φ) → (∃x x = A → ψ))
9	1, 8	syl5com 26	. 2 ⊢ (A ∈ V → (∀x(x = A → φ) → ψ))
10	4	biimprcd 216	. . 3 ⊢ (ψ → (x = A → φ))
11	3, 10	alrimi 1765	. 2 ⊢ (ψ → ∀x(x = A → φ))
12	9, 11	impbid1 194	1 ⊢ (A ∈ V → (∀x(x = A → φ) ↔ ψ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710

This proof depends on 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 proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is used by:  ceqsal  2885  sbc6g  3072  uniiunlem  3354