Theorem ceqsal

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 . (Revised by Wolf Lammen, 23-Jan-2025)

	Ref	Expression
Hypotheses	ceqsal.1	⊢ Ⅎ 𝑥 𝜓
	ceqsal.2	⊢ 𝐴 ∈ V
	ceqsal.3	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	ceqsal	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		ceqsal.1	⊢ Ⅎ 𝑥 𝜓
2		ceqsal.2	⊢ 𝐴 ∈ V
3		ceqsal.3	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
4	1	19.23	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) )
5	3	pm5.74i	⊢ ( ( 𝑥 = 𝐴 → 𝜑 ) ↔ ( 𝑥 = 𝐴 → 𝜓 ) )
6	5	albii	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) )
7	2	isseti	⊢ ∃ 𝑥 𝑥 = 𝐴
8	7	a1bi	⊢ ( 𝜓 ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) )
9	4 6 8	3bitr4i	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 )