Theorem ceqsexg

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004)

	Ref	Expression
Hypotheses	ceqsexg.1	⊢ Ⅎ 𝑥 𝜓
	ceqsexg.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	ceqsexg	⊢ ( 𝐴 ∈ 𝑉 → ( ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		ceqsexg.1	⊢ Ⅎ 𝑥 𝜓
2		ceqsexg.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3		nfe1	⊢ Ⅎ 𝑥 ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 )
4	3 1	nfbi	⊢ Ⅎ 𝑥 ( ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) ↔ 𝜓 )
5		ceqex	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) ) )
6	5 2	bibi12d	⊢ ( 𝑥 = 𝐴 → ( ( 𝜑 ↔ 𝜑 ) ↔ ( ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) ↔ 𝜓 ) ) )
7		biid	⊢ ( 𝜑 ↔ 𝜑 )
8	4 6 7	vtoclg1f	⊢ ( 𝐴 ∈ 𝑉 → ( ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) ↔ 𝜓 ) )