Theorem dveeq2 1693

Description: Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)

Assertion

Ref	Expression
dveeq2	⊢ (¬ ∀x x = y → (z = y → ∀x z = y))

Distinct variable group:   x,z

Proof of Theorem dveeq2

Step	Hyp	Ref	Expression
1		ax-i12 1395	. . . . 5 ⊢ (∀x x = z ∨ (∀x x = y ∨ ∀x(z = y → ∀x z = y)))
2		orcom 646	. . . . . 6 ⊢ ((∀x x = y ∨ ∀x(z = y → ∀x z = y)) ↔ (∀x(z = y → ∀x z = y) ∨ ∀x x = y))
3	2	orbi2i 678	. . . . 5 ⊢ ((∀x x = z ∨ (∀x x = y ∨ ∀x(z = y → ∀x z = y))) ↔ (∀x x = z ∨ (∀x(z = y → ∀x z = y) ∨ ∀x x = y)))
4	1, 3	mpbi 133	. . . 4 ⊢ (∀x x = z ∨ (∀x(z = y → ∀x z = y) ∨ ∀x x = y))
5		orass 683	. . . 4 ⊢ (((∀x x = z ∨ ∀x(z = y → ∀x z = y)) ∨ ∀x x = y) ↔ (∀x x = z ∨ (∀x(z = y → ∀x z = y) ∨ ∀x x = y)))
6	4, 5	mpbir 134	. . 3 ⊢ ((∀x x = z ∨ ∀x(z = y → ∀x z = y)) ∨ ∀x x = y)
7		orel2 644	. . 3 ⊢ (¬ ∀x x = y → (((∀x x = z ∨ ∀x(z = y → ∀x z = y)) ∨ ∀x x = y) → (∀x x = z ∨ ∀x(z = y → ∀x z = y))))
8	6, 7	mpi 15	. 2 ⊢ (¬ ∀x x = y → (∀x x = z ∨ ∀x(z = y → ∀x z = y)))
9		ax16 1691	. . 3 ⊢ (∀x x = z → (z = y → ∀x z = y))
10		sp 1398	. . 3 ⊢ (∀x(z = y → ∀x z = y) → (z = y → ∀x z = y))
11	9, 10	jaoi 635	. 2 ⊢ ((∀x x = z ∨ ∀x(z = y → ∀x z = y)) → (z = y → ∀x z = y))
12	8, 11	syl 14	1 ⊢ (¬ ∀x x = y → (z = y → ∀x z = y))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 628  ∀wal 1240

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643

This theorem is referenced by:  nd5  1696  ax11v2  1698  dveeq1  1892