Theorem spcegf

Description: Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997)

	Ref	Expression
Hypotheses	spcgf.1	⊢ Ⅎ 𝑥 𝐴
	spcgf.2	⊢ Ⅎ 𝑥 𝜓
	spcgf.3	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	spcegf	⊢ ( 𝐴 ∈ 𝑉 → ( 𝜓 → ∃ 𝑥 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		spcgf.1	⊢ Ⅎ 𝑥 𝐴
2		spcgf.2	⊢ Ⅎ 𝑥 𝜓
3		spcgf.3	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
4	2	nfn	⊢ Ⅎ 𝑥 ¬ 𝜓
5	3	notbid	⊢ ( 𝑥 = 𝐴 → ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
6	1 4 5	spcgf	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ¬ 𝜑 → ¬ 𝜓 ) )
7	6	con2d	⊢ ( 𝐴 ∈ 𝑉 → ( 𝜓 → ¬ ∀ 𝑥 ¬ 𝜑 ) )
8		df-ex	⊢ ( ∃ 𝑥 𝜑 ↔ ¬ ∀ 𝑥 ¬ 𝜑 )
9	7 8	syl6ibr	⊢ ( 𝐴 ∈ 𝑉 → ( 𝜓 → ∃ 𝑥 𝜑 ) )