Theorem ceqsal 3531

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 3529	. 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 208  ∀wal 1531   = wceq 1533  Ⅎwnf 1780   ∈ wcel 2110  Vcvv 3494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-3an 1085  df-ex 1777  df-nf 1781  df-cleq 2814  df-clel 2893

This theorem is referenced by:  ceqsalv  3532  aomclem6  39657