Theorem ceqsal 3507

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.) Avoid df-clab 2706. (Revised by Wolf Lammen, 23-Jan-2025.)

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.1	. . 3 ⊢ Ⅎ𝑥𝜓
2	1	19.23 2200	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓))
3		ceqsal.3	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	3	pm5.74i 271	. . 3 ⊢ ((𝑥 = 𝐴 → 𝜑) ↔ (𝑥 = 𝐴 → 𝜓))
5	4	albii 1814	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜓))
6		ceqsal.2	. . . 4 ⊢ 𝐴 ∈ V
7	6	isseti 3487	. . 3 ⊢ ∃𝑥 𝑥 = 𝐴
8	7	a1bi 362	. 2 ⊢ (𝜓 ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓))
9	2, 5, 8	3bitr4i 303	1 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1532   = wceq 1534  ∃wex 1774  Ⅎwnf 1778   ∈ wcel 2099  Vcvv 3471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-12 2167

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-nf 1779  df-clel 2806

This theorem is referenced by:  ceqsalvOLD  3510  ceqsex  3521  ralxp3f  8142  aomclem6  42483