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Metamath Proof Explorer

Theorem spc2egv

Description: Existential specialization with two quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995)

Ref Expression
Hypothesis spc2egv.1 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
Assertion spc2egv ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝜓 → ∃ 𝑥 ∃ 𝑦 𝜑 ) )

Proof

Step Hyp Ref Expression
1 spc2egv.1 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
2 elisset ⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 )
3 elisset ⊢ ( 𝐵 ∈ 𝑊 → ∃ 𝑦 𝑦 = 𝐵 )
4 2 3 anim12i ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) )
5 exdistrv ⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) )
6 4 5 sylibr ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) )
7 1 biimprcd ⊢ ( 𝜓 → ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → 𝜑 ) )
8 7 2eximdv ⊢ ( 𝜓 → ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ∃ 𝑥 ∃ 𝑦 𝜑 ) )
9 6 8 syl5com ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝜓 → ∃ 𝑥 ∃ 𝑦 𝜑 ) )