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Theorem ax16 1710
Description: Theorem showing that ax-16 1711 is redundant if ax-17 1435 is included in the axiom system. The important part of the proof is provided by aev 1709.

See ax16ALT 1755 for an alternate proof that does not require ax-10 1412 or ax-12 1418.

This theorem should not be referenced in any proof. Instead, use ax-16 1711 below so that theorems needing ax-16 1711 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 1709 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑧)
2 ax-17 1435 . . . 4 (𝜑 → ∀𝑧𝜑)
3 sbequ12 1670 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
43biimpcd 152 . . . 4 (𝜑 → (𝑥 = 𝑧 → [𝑧 / 𝑥]𝜑))
52, 4alimdh 1372 . . 3 (𝜑 → (∀𝑧 𝑥 = 𝑧 → ∀𝑧[𝑧 / 𝑥]𝜑))
62hbsb3 1705 . . . 4 ([𝑧 / 𝑥]𝜑 → ∀𝑥[𝑧 / 𝑥]𝜑)
7 stdpc7 1669 . . . 4 (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝜑𝜑))
86, 2, 7cbv3h 1647 . . 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 1257  [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662
This theorem is referenced by:  dveeq2  1712  dveeq2or  1713  a16g  1760  exists2  2013
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