Theorem ceqsal 2885

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)

Hypotheses

Ref	Expression
ceqsal.1	⊢ Ⅎxψ
ceqsal.2	⊢ A ∈ V
ceqsal.3	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
ceqsal	⊢ (∀x(x = A → φ) ↔ ψ)

Distinct variable group:   x,A

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

Proof of Theorem ceqsal

Step	Hyp	Ref	Expression
1		ceqsal.2	. 2 ⊢ A ∈ V
2		ceqsal.1	. . 3 ⊢ Ⅎxψ
3		ceqsal.3	. . 3 ⊢ (x = A → (φ ↔ ψ))
4	2, 3	ceqsalg 2884	. 2 ⊢ (A ∈ V → (∀x(x = A → φ) ↔ ψ))
5	1, 4	ax-mp 5	1 ⊢ (∀x(x = A → φ) ↔ ψ)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Vcvv 2860

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-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  ceqsalv  2886