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 )