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Theorem axc16ALT 2394
Description: Alternate proof of axc16 2173, shorter but requiring ax-10 2059, ax-11 2074, ax-13 2282 and using df-nf 1750 and df-sb 1938. (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 2149 . 2 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
2 ax-5 1879 . . 3 (𝜑 → ∀𝑧𝜑)
32hbsb3 2392 . 2 ([𝑧 / 𝑥]𝜑 → ∀𝑥[𝑧 / 𝑥]𝜑)
41, 3axc16i 2353 1 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1521  [wsb 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-sb 1938
This theorem is referenced by:  axc16gALT  2395
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