Theorem spcimgft

Description: Closed theorem form of spcimgf . (Contributed by Wolf Lammen, 28-Jul-2025)

	Ref	Expression
Assertion	spcimgft	⊢ ( ( ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) ) ) → ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 𝜑 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elissetv	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑦 𝑦 = 𝐴 )
2		cbvexeqsetf	⊢ ( Ⅎ 𝑥 𝐴 → ( ∃ 𝑥 𝑥 = 𝐴 ↔ ∃ 𝑦 𝑦 = 𝐴 ) )
3	1 2	imbitrrid	⊢ ( Ⅎ 𝑥 𝐴 → ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 ) )
4		pm2.04	⊢ ( ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → ( 𝑥 = 𝐴 → 𝜓 ) ) )
5	4	al2imi	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) ) → ( ∀ 𝑥 𝜑 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ) )
6		19.23t	⊢ ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) ) )
7	6	biimpd	⊢ ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) → ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) ) )
8	5 7	sylan9r	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) ) ) → ( ∀ 𝑥 𝜑 → ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) ) )
9	8	com23	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) ) ) → ( ∃ 𝑥 𝑥 = 𝐴 → ( ∀ 𝑥 𝜑 → 𝜓 ) ) )
10	3 9	sylan9	⊢ ( ( Ⅎ 𝑥 𝐴 ∧ ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) ) ) ) → ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 𝜑 → 𝜓 ) ) )
11	10	anassrs	⊢ ( ( ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) ) ) → ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 𝜑 → 𝜓 ) ) )