Theorem dveeq2 1678

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 1379	. . . . 5 ⊢ (∀x x = z ∨ (∀x x = y ∨ ∀x(z = y → ∀x z = y)))
2		orcom 634	. . . . . 6 ⊢ ((∀x x = y ∨ ∀x(z = y → ∀x z = y)) ↔ (∀x(z = y → ∀x z = y) ∨ ∀x x = y))
3	2	orbi2i 666	. . . . 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 671	. . . 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 632	. . 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 1676	. . 3 ⊢ (∀x x = z → (z = y → ∀x z = y))
10		sp 1382	. . 3 ⊢ (∀x(z = y → ∀x z = y) → (z = y → ∀x z = y))
11	9, 10	jaoi 623	. 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 616  ∀wal 1226

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 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409

This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628

This theorem is referenced by:  nd5  1681  ax11v2  1683  dveeq1  1877