Theorem vtocl2OLD

Description: Obsolete proof of vtocl2 as of 23-Aug-2023. (Contributed by NM, 26-Jul-1995) (Proof shortened by Andrew Salmon, 8-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	vtocl2.1	⊢ 𝐴 ∈ V
	vtocl2.2	⊢ 𝐵 ∈ V
	vtocl2.3	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
	vtocl2.4	⊢ 𝜑
Assertion	vtocl2OLD	⊢ 𝜓

Proof

Step	Hyp	Ref	Expression
1		vtocl2.1	⊢ 𝐴 ∈ V
2		vtocl2.2	⊢ 𝐵 ∈ V
3		vtocl2.3	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
4		vtocl2.4	⊢ 𝜑
5	1	isseti	⊢ ∃ 𝑥 𝑥 = 𝐴
6	2	isseti	⊢ ∃ 𝑦 𝑦 = 𝐵
7		exdistrv	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) )
8	3	biimpd	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 → 𝜓 ) )
9	8	2eximi	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ∃ 𝑥 ∃ 𝑦 ( 𝜑 → 𝜓 ) )
10	7 9	sylbir	⊢ ( ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) → ∃ 𝑥 ∃ 𝑦 ( 𝜑 → 𝜓 ) )
11	5 6 10	mp2an	⊢ ∃ 𝑥 ∃ 𝑦 ( 𝜑 → 𝜓 )
12		19.36v	⊢ ( ∃ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑦 𝜑 → 𝜓 ) )
13	12	exbii	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 → 𝜓 ) ↔ ∃ 𝑥 ( ∀ 𝑦 𝜑 → 𝜓 ) )
14	11 13	mpbi	⊢ ∃ 𝑥 ( ∀ 𝑦 𝜑 → 𝜓 )
15	14	19.36iv	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 → 𝜓 )
16	4	ax-gen	⊢ ∀ 𝑦 𝜑
17	15 16	mpg	⊢ 𝜓