Theorem dvelimc 2949

Description: Version of dvelim 2482 for classes. Usage of this theorem is discouraged because it depends on ax-13 2403. (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 1824	. . 3 ⊢ Ⅎ𝑥⊤
2		nftru 1824	. . 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 2948	. 2 ⊢ (⊤ → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵))
10	9	mptru 1567	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1558   = wceq 1560  ⊤wtru 1561  Ⅎwnfc 2909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-13 2403  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-nf 1804  df-cleq 2754  df-clel 2837  df-nfc 2911

This theorem is referenced by: (None)