Theorem dvelimc 2510

Description: Version of dvelim 2016 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.)

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 1554	. . 3 ⊢ Ⅎx ⊤
2		nftru 1554	. . 3 ⊢ Ⅎz ⊤
3		dvelimc.1	. . . 4 ⊢ ℲxA
4	3	a1i 10	. . 3 ⊢ ( ⊤ → ℲxA)
5		dvelimc.2	. . . 4 ⊢ ℲzB
6	5	a1i 10	. . 3 ⊢ ( ⊤ → ℲzB)
7		dvelimc.3	. . . 4 ⊢ (z = y → A = B)
8	7	a1i 10	. . 3 ⊢ ( ⊤ → (z = y → A = B))
9	1, 2, 4, 6, 8	dvelimdc 2509	. 2 ⊢ ( ⊤ → (¬ ∀x x = y → ℲxB))
10	9	trud 1323	1 ⊢ (¬ ∀x x = y → ℲxB)

Syntax hints:  ¬ wn 3   → wi 4   ⊤ wtru 1316  ∀wal 1540   = wceq 1642  Ⅎwnfc 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478

This theorem is referenced by:  nfcvf  2511