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Theorem dveeq2 1769
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
StepHypRef Expression
1 ax-i12 1468 . . . . 5 (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)))
2 orcom 700 . . . . . 6 ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) ↔ (∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) ∨ ∀𝑥 𝑥 = 𝑦))
32orbi2i 734 . . . . 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)))
41, 3mpbi 144 . . . 4 (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) ∨ ∀𝑥 𝑥 = 𝑦))
5 orass 739 . . . 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)))
64, 5mpbir 145 . . 3 ((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) ∨ ∀𝑥 𝑥 = 𝑦)
7 orel2 698 . . 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))))
86, 7mpi 15 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)))
9 ax16 1767 . . 3 (∀𝑥 𝑥 = 𝑧 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
10 sp 1471 . . 3 (∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
119, 10jaoi 688 . 2 ((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
128, 11syl 14 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 680  wal 1312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719
This theorem is referenced by:  nd5  1772  ax11v2  1774  dveeq1  1970
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