Theorem cgsex2g

Description: Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995)

	Ref	Expression
Hypotheses	cgsex2g.1	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → 𝜒 )
	cgsex2g.2	⊢ ( 𝜒 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cgsex2g	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∃ 𝑥 ∃ 𝑦 ( 𝜒 ∧ 𝜑 ) ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		cgsex2g.1	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → 𝜒 )
2		cgsex2g.2	⊢ ( 𝜒 → ( 𝜑 ↔ 𝜓 ) )
3	2	biimpa	⊢ ( ( 𝜒 ∧ 𝜑 ) → 𝜓 )
4	3	exlimivv	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜒 ∧ 𝜑 ) → 𝜓 )
5		elisset	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 )
6		elisset	⊢ ( 𝐵 ∈ 𝑊 → ∃ 𝑦 𝑦 = 𝐵 )
7	5 6	anim12i	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) )
8		exdistrv	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) )
9	7 8	sylibr	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) )
10	1	2eximi	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ∃ 𝑥 ∃ 𝑦 𝜒 )
11	9 10	syl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ∃ 𝑥 ∃ 𝑦 𝜒 )
12	2	biimprcd	⊢ ( 𝜓 → ( 𝜒 → 𝜑 ) )
13	12	ancld	⊢ ( 𝜓 → ( 𝜒 → ( 𝜒 ∧ 𝜑 ) ) )
14	13	2eximdv	⊢ ( 𝜓 → ( ∃ 𝑥 ∃ 𝑦 𝜒 → ∃ 𝑥 ∃ 𝑦 ( 𝜒 ∧ 𝜑 ) ) )
15	11 14	syl5com	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝜓 → ∃ 𝑥 ∃ 𝑦 ( 𝜒 ∧ 𝜑 ) ) )
16	4 15	impbid2	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∃ 𝑥 ∃ 𝑦 ( 𝜒 ∧ 𝜑 ) ↔ 𝜓 ) )