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Theorem axc16gALT 2493
Description: Alternate proof of axc16g 2258 that uses df-sb 2063 and requires ax-10 2139, ax-11 2155, ax-13 2375. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc16gALT (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem axc16gALT
StepHypRef Expression
1 aev 2055 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑥)
2 axc16ALT 2492 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
3 biidd 262 . . . 4 (∀𝑧 𝑧 = 𝑥 → (𝜑𝜑))
43dral1 2442 . . 3 (∀𝑧 𝑧 = 𝑥 → (∀𝑧𝜑 ↔ ∀𝑥𝜑))
54biimprd 248 . 2 (∀𝑧 𝑧 = 𝑥 → (∀𝑥𝜑 → ∀𝑧𝜑))
61, 2, 5sylsyld 61 1 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063
This theorem is referenced by: (None)
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