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Theorem hbsb2 1792
Description: Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
hbsb2 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))

Proof of Theorem hbsb2
StepHypRef Expression
1 sb4 1788 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 sb2 1725 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
32a5i 1507 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥[𝑦 / 𝑥]𝜑)
41, 3syl6 33 1 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1314  [wsb 1720
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-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721
This theorem is referenced by: (None)
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