Theorem ceqsalv

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 ax-12 . (Revised by SN, 8-Sep-2024)

	Ref	Expression
Hypotheses	ceqsalv.1	⊢ 𝐴 ∈ V
	ceqsalv.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	ceqsalv	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		ceqsalv.1	⊢ 𝐴 ∈ V
2		ceqsalv.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3		19.23v	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) )
4	2	pm5.74i	⊢ ( ( 𝑥 = 𝐴 → 𝜑 ) ↔ ( 𝑥 = 𝐴 → 𝜓 ) )
5	4	albii	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) )
6	1	isseti	⊢ ∃ 𝑥 𝑥 = 𝐴
7	6	a1bi	⊢ ( 𝜓 ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) )
8	3 5 7	3bitr4i	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 )