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Theorem dveeq2or 1739
Description: Quantifier introduction when one pair of variables is distinct. Like dveeq2 1738 but connecting 𝑥𝑥 = 𝑦 by a disjunction rather than negation and implication makes the theorem stronger in intuitionistic logic. (Contributed by Jim Kingdon, 1-Feb-2018.)
Assertion
Ref Expression
dveeq2or (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥 𝑧 = 𝑦)
Distinct variable group:   𝑥,𝑧

Proof of Theorem dveeq2or
StepHypRef Expression
1 ax-i12 1439 . . . . . 6 (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)))
2 orass 717 . . . . . 6 (((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦) ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) ↔ (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))))
31, 2mpbir 144 . . . . 5 ((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦) ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
4 pm1.4 679 . . . . . 6 ((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦) → (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧))
54orim1i 710 . . . . 5 (((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥 𝑥 = 𝑦) ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) → ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧) ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)))
63, 5ax-mp 7 . . . 4 ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥 𝑥 = 𝑧) ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
7 orass 717 . . . 4 (((∀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, 7mpbi 143 . . 3 (∀𝑥 𝑥 = 𝑦 ∨ (∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)))
9 ax16 1736 . . . . . 6 (∀𝑥 𝑥 = 𝑧 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
109a5i 1476 . . . . 5 (∀𝑥 𝑥 = 𝑧 → ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
11 id 19 . . . . 5 (∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) → ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
1210, 11jaoi 669 . . . 4 ((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) → ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
1312orim2i 711 . . 3 ((∀𝑥 𝑥 = 𝑦 ∨ (∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))) → (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)))
148, 13ax-mp 7 . 2 (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
15 df-nf 1391 . . . 4 (Ⅎ𝑥 𝑧 = 𝑦 ↔ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
1615biimpri 131 . . 3 (∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) → Ⅎ𝑥 𝑧 = 𝑦)
1716orim2i 711 . 2 ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) → (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥 𝑧 = 𝑦))
1814, 17ax-mp 7 1 (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥 𝑧 = 𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 662  wal 1283  wnf 1390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688
This theorem is referenced by:  equs5or  1753  sbal1yz  1920  copsexg  4035
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