Theorem cgsexg

Description: Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007)

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

Proof

Step	Hyp	Ref	Expression
1		cgsexg.1	⊢ ( 𝑥 = 𝐴 → 𝜒 )
2		cgsexg.2	⊢ ( 𝜒 → ( 𝜑 ↔ 𝜓 ) )
3	2	biimpa	⊢ ( ( 𝜒 ∧ 𝜑 ) → 𝜓 )
4	3	exlimiv	⊢ ( ∃ 𝑥 ( 𝜒 ∧ 𝜑 ) → 𝜓 )
5		elisset	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 )
6	1	eximi	⊢ ( ∃ 𝑥 𝑥 = 𝐴 → ∃ 𝑥 𝜒 )
7	5 6	syl	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝜒 )
8	2	biimprcd	⊢ ( 𝜓 → ( 𝜒 → 𝜑 ) )
9	8	ancld	⊢ ( 𝜓 → ( 𝜒 → ( 𝜒 ∧ 𝜑 ) ) )
10	9	eximdv	⊢ ( 𝜓 → ( ∃ 𝑥 𝜒 → ∃ 𝑥 ( 𝜒 ∧ 𝜑 ) ) )
11	7 10	syl5com	⊢ ( 𝐴 ∈ 𝑉 → ( 𝜓 → ∃ 𝑥 ( 𝜒 ∧ 𝜑 ) ) )
12	4 11	impbid2	⊢ ( 𝐴 ∈ 𝑉 → ( ∃ 𝑥 ( 𝜒 ∧ 𝜑 ) ↔ 𝜓 ) )