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Metamath Proof Explorer

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 ) ) )