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Theorem axc16g 2312
Description: Generalization of axc16 2313. Use the latter when sufficient. This proof only requires, on top of { ax-1 6-- ax-7 2105 }, theorem ax12v 2212. (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 2352, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 7-Jul-2021.) Shorten axc11rv 2317. (Revised by Wolf Lammen, 11-Oct-2021.)
Assertion
Ref Expression
axc16g (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem axc16g
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 aevlem 2148 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑤)
2 ax12v 2212 . . . 4 (𝑧 = 𝑤 → (𝜑 → ∀𝑧(𝑧 = 𝑤𝜑)))
32sps 2217 . . 3 (∀𝑧 𝑧 = 𝑤 → (𝜑 → ∀𝑧(𝑧 = 𝑤𝜑)))
4 pm2.27 42 . . . 4 (𝑧 = 𝑤 → ((𝑧 = 𝑤𝜑) → 𝜑))
54al2imi 1910 . . 3 (∀𝑧 𝑧 = 𝑤 → (∀𝑧(𝑧 = 𝑤𝜑) → ∀𝑧𝜑))
63, 5syld 47 . 2 (∀𝑧 𝑧 = 𝑤 → (𝜑 → ∀𝑧𝜑))
71, 6syl 17 1 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-12 2211
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1875
This theorem is referenced by:  axc16  2313  axc16gb  2314  axc16nf  2315  axc11v  2316  axc11rv  2317  axc16nfALT  2417
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