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Theorem axc16ALT 2482
Description: Alternate proof of axc16 2244, shorter but requiring ax-10 2129, ax-11 2146, ax-13 2365 and using df-nf 1778 and df-sb 2060. (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 2235 . 2 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
2 ax-5 1905 . . 3 (𝜑 → ∀𝑧𝜑)
32hbsb3 2480 . 2 ([𝑧 / 𝑥]𝜑 → ∀𝑥[𝑧 / 𝑥]𝜑)
41, 3axc16i 2429 1 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531  [wsb 2059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-11 2146  ax-12 2163  ax-13 2365
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-nf 1778  df-sb 2060
This theorem is referenced by:  axc16gALT  2483
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