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Theorem equs5 1751
Description: Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
equs5 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))

Proof of Theorem equs5
StepHypRef Expression
1 hbnae 1650 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦)
2 hba1 1474 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥𝑥(𝑥 = 𝑦𝜑))
3 ax11o 1744 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
43impd 251 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
51, 2, 4exlimdh 1528 1 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wal 1283  wex 1422
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-in1 577  ax-in2 578  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-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687
This theorem is referenced by:  sb3  1753  sb4  1754
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