Theorem spcimgft

Description: A closed version of spcimgf . (Contributed by Mario Carneiro, 4-Jan-2017)

	Ref	Expression
Hypotheses	spcimgft.1	⊢ Ⅎ 𝑥 𝜓
	spcimgft.2	⊢ Ⅎ 𝑥 𝐴
Assertion	spcimgft	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) ) → ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 𝜑 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		spcimgft.1	⊢ Ⅎ 𝑥 𝜓
2		spcimgft.2	⊢ Ⅎ 𝑥 𝐴
3		elex	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ∈ V )
4	2	issetf	⊢ ( 𝐴 ∈ V ↔ ∃ 𝑥 𝑥 = 𝐴 )
5		exim	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) ) → ( ∃ 𝑥 𝑥 = 𝐴 → ∃ 𝑥 ( 𝜑 → 𝜓 ) ) )
6	4 5	syl5bi	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) ) → ( 𝐴 ∈ V → ∃ 𝑥 ( 𝜑 → 𝜓 ) ) )
7	1	19.36	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 𝜑 → 𝜓 ) )
8	6 7	syl6ib	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) ) → ( 𝐴 ∈ V → ( ∀ 𝑥 𝜑 → 𝜓 ) ) )
9	3 8	syl5	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) ) → ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 𝜑 → 𝜓 ) ) )