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Theorem axc16ALT 2507
 Description: Alternate proof of axc16 2259, shorter but requiring ax-10 2142, ax-11 2158, ax-13 2379 and using df-nf 1786 and df-sb 2070. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc16ALT (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem axc16ALT
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sbequ12 2250 . 2 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
2 ax-5 1911 . . 3 (𝜑 → ∀𝑧𝜑)
32hbsb3 2505 . 2 ([𝑧 / 𝑥]𝜑 → ∀𝑥[𝑧 / 𝑥]𝜑)
41, 3axc16i 2447 1 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070 This theorem is referenced by:  axc16gALT  2508
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