Theorem spcimgft 3514

Description: Closed theorem form of spcimgf 3517. (Contributed by Wolf Lammen, 28-Jul-2025.)

Assertion

Ref	Expression
spcimgft	⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓))) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)))

Proof of Theorem spcimgft

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elissetv 2843	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦 𝑦 = 𝐴)
2		cbvexeqsetf 3469	. . . 4 ⊢ (Ⅎ𝑥𝐴 → (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴))
3	1, 2	imbitrrid 249	. . 3 ⊢ (Ⅎ𝑥𝐴 → (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴))
4		pm2.04 91	. . . . . 6 ⊢ ((𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝜑 → (𝑥 = 𝐴 → 𝜓)))
5	4	al2imi 1844	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝜓)))
6		19.23t 2245	. . . . . 6 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓)))
7	6	biimpd 232	. . . . 5 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴 → 𝜓) → (∃𝑥 𝑥 = 𝐴 → 𝜓)))
8	5, 7	sylan9r 517	. . . 4 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → (∃𝑥 𝑥 = 𝐴 → 𝜓)))
9	8	com23 87	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓))) → (∃𝑥 𝑥 = 𝐴 → (∀𝑥𝜑 → 𝜓)))
10	3, 9	sylan9 516	. 2 ⊢ ((Ⅎ𝑥𝐴 ∧ (Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)))) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)))
11	10	anassrs 472	1 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓))) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400  ∀wal 1567   = wceq 1569  ∃wex 1808  Ⅎwnf 1812   ∈ wcel 2142  Ⅎwnfc 2909

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1809  df-nf 1813  df-cleq 2754  df-clel 2837  df-nfc 2911

This theorem is used by:  spcimgfi1  3515  vtoclgft  3519