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Theorem axc16g 2281
Description: Generalization of axc16 2282. Use the latter when sufficient. This proof only requires, on top of { ax-1 6-- ax-7 2090 }, theorem ax12v 2197. (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 2391, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 7-Jul-2021.) Shorten axc11rv 2286. (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 2132 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑤)
2 ax12v 2197 . . . 4 (𝑧 = 𝑤 → (𝜑 → ∀𝑧(𝑧 = 𝑤𝜑)))
32sps 2202 . . 3 (∀𝑧 𝑧 = 𝑤 → (𝜑 → ∀𝑧(𝑧 = 𝑤𝜑)))
4 pm2.27 42 . . . 4 (𝑧 = 𝑤 → ((𝑧 = 𝑤𝜑) → 𝜑))
54al2imi 1892 . . 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 1630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-12 2196
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854
This theorem is referenced by:  axc16  2282  axc16gb  2283  axc16nf  2284  axc11v  2285  axc11rv  2286  axc16nfALT  2463
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