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Theorem axc16ALT 2365
Description: Alternate proof of axc16 2131, shorter but requiring ax-10 2016, ax-11 2031, ax-13 2245 and using df-nf 1707 and df-sb 1878. (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 2108 . 2 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
2 ax-5 1836 . . 3 (𝜑 → ∀𝑧𝜑)
32hbsb3 2363 . 2 ([𝑧 / 𝑥]𝜑 → ∀𝑥[𝑧 / 𝑥]𝜑)
41, 3axc16i 2321 1 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478  [wsb 1877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-sb 1878
This theorem is referenced by:  axc16gALT  2366
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