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Metamath Proof Explorer

Theorem spc2ev

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

Ref Expression
Hypotheses spc2ev.1 ⊢ 𝐴 ∈ V
spc2ev.2 ⊢ 𝐵 ∈ V
spc2ev.3 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
Assertion spc2ev ( 𝜓 → ∃ 𝑥 ∃ 𝑦 𝜑 )

Proof

Step Hyp Ref Expression
1 spc2ev.1 ⊢ 𝐴 ∈ V
2 spc2ev.2 ⊢ 𝐵 ∈ V
3 spc2ev.3 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
4 3 spc2egv ⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝜓 → ∃ 𝑥 ∃ 𝑦 𝜑 ) )
5 1 2 4 mp2an ⊢ ( 𝜓 → ∃ 𝑥 ∃ 𝑦 𝜑 )