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Metamath Proof Explorer

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 ) )