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Theorem ax16 1694
 Description: Theorem showing that ax-16 1695 is redundant if ax-17 1419 is included in the axiom system. The important part of the proof is provided by aev 1693. See ax16ALT 1739 for an alternate proof that does not require ax-10 1396 or ax-12 1402. This theorem should not be referenced in any proof. Instead, use ax-16 1695 below so that theorems needing ax-16 1695 can be more easily identified. (Contributed by NM, 8-Nov-2006.)
Assertion
Ref Expression
ax16 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax16
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 aev 1693 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑧)
2 ax-17 1419 . . . 4 (𝜑 → ∀𝑧𝜑)
3 sbequ12 1654 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
43biimpcd 148 . . . 4 (𝜑 → (𝑥 = 𝑧 → [𝑧 / 𝑥]𝜑))
52, 4alimdh 1356 . . 3 (𝜑 → (∀𝑧 𝑥 = 𝑧 → ∀𝑧[𝑧 / 𝑥]𝜑))
62hbsb3 1689 . . . 4 ([𝑧 / 𝑥]𝜑 → ∀𝑥[𝑧 / 𝑥]𝜑)
7 stdpc7 1653 . . . 4 (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝜑𝜑))
86, 2, 7cbv3h 1631 . . 3 (∀𝑧[𝑧 / 𝑥]𝜑 → ∀𝑥𝜑)
95, 8syl6com 31 . 2 (∀𝑧 𝑥 = 𝑧 → (𝜑 → ∀𝑥𝜑))
101, 9syl 14 1 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1241  [wsb 1645 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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646 This theorem is referenced by:  dveeq2  1696  dveeq2or  1697  a16g  1744  exists2  1997
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