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Theorem ax16 1769
Description: Theorem showing that ax-16 1770 is redundant if ax-17 1491 is included in the axiom system. The important part of the proof is provided by aev 1768.

See ax16ALT 1815 for an alternate proof that does not require ax-10 1468 or ax-12 1474.

This theorem should not be referenced in any proof. Instead, use ax-16 1770 below so that theorems needing ax-16 1770 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 1768 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑧)
2 ax-17 1491 . . . 4 (𝜑 → ∀𝑧𝜑)
3 sbequ12 1729 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
43biimpcd 158 . . . 4 (𝜑 → (𝑥 = 𝑧 → [𝑧 / 𝑥]𝜑))
52, 4alimdh 1428 . . 3 (𝜑 → (∀𝑧 𝑥 = 𝑧 → ∀𝑧[𝑧 / 𝑥]𝜑))
62hbsb3 1764 . . . 4 ([𝑧 / 𝑥]𝜑 → ∀𝑥[𝑧 / 𝑥]𝜑)
7 stdpc7 1728 . . . 4 (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝜑𝜑))
86, 2, 7cbv3h 1706 . . 3 (∀𝑧[𝑧 / 𝑥]𝜑 → ∀𝑥𝜑)
95, 8syl6com 35 . 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 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-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-nf 1422  df-sb 1721
This theorem is referenced by:  dveeq2  1771  dveeq2or  1772  a16g  1820  exists2  2074
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