Theorem dvelimc 2925

Description: Version of dvelim 2450 for classes. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dvelimc.1	⊢ Ⅎ𝑥𝐴
dvelimc.2	⊢ Ⅎ𝑧𝐵
dvelimc.3	⊢ (𝑧 = 𝑦 → 𝐴 = 𝐵)

Assertion

Ref	Expression
dvelimc	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵)

Proof of Theorem dvelimc

Step	Hyp	Ref	Expression
1		nftru 1812	. . 3 ⊢ Ⅎ𝑥⊤
2		nftru 1812	. . 3 ⊢ Ⅎ𝑧⊤
3		dvelimc.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4	3	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
5		dvelimc.2	. . . 4 ⊢ Ⅎ𝑧𝐵
6	5	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑧𝐵)
7		dvelimc.3	. . . 4 ⊢ (𝑧 = 𝑦 → 𝐴 = 𝐵)
8	7	a1i 11	. . 3 ⊢ (⊤ → (𝑧 = 𝑦 → 𝐴 = 𝐵))
9	1, 2, 4, 6, 8	dvelimdc 2924	. 2 ⊢ (⊤ → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵))
10	9	mptru 1550	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1541   = wceq 1543  ⊤wtru 1544  Ⅎwnfc 2877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-13 2371  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-cleq 2728  df-clel 2809  df-nfc 2879

This theorem is referenced by: (None)