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Theorem axc16gOLD 2160
Description: Obsolete proof of axc16g 2133 as of 11-Oct-2021. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 18-Feb-2018.) Remove dependency on ax-13 2245, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 7-Jul-2021.) Shorten axc11rv 2138. (Revised by Wolf Lammen, 11-Oct-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
axc16gOLD (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem axc16gOLD
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 aevlem 1980 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑤)
2 ax-5 1838 . 2 (𝜑 → ∀𝑤𝜑)
3 axc11rvOLD 2139 . 2 (∀𝑧 𝑧 = 𝑤 → (∀𝑤𝜑 → ∀𝑧𝜑))
41, 2, 3syl2im 40 1 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-12 2046
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1704
This theorem is referenced by: (None)
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