Theorem spcimegf

Description: Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017)

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

Proof

Step	Hyp	Ref	Expression
1		spcimgf.1	⊢ Ⅎ 𝑥 𝐴
2		spcimgf.2	⊢ Ⅎ 𝑥 𝜓
3		spcimegf.3	⊢ ( 𝑥 = 𝐴 → ( 𝜓 → 𝜑 ) )
4	2	nfn	⊢ Ⅎ 𝑥 ¬ 𝜓
5	3	con3d	⊢ ( 𝑥 = 𝐴 → ( ¬ 𝜑 → ¬ 𝜓 ) )
6	1 4 5	spcimgf	⊢ ( 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 ) )