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Theorem axc16g 2119
 Description: Generalization of axc16 2120. Use the latter when sufficient. This proof only requires, on top of { ax-1 6-- ax-7 1922 }, theorem ax12v 2035. (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 2234, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 7-Jul-2021.) Shorten axc11rv 2124. (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 1968 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑤)
2 ax12v2 2036 . . . 4 (𝑧 = 𝑤 → (𝜑 → ∀𝑧(𝑧 = 𝑤𝜑)))
32sps 2043 . . 3 (∀𝑧 𝑧 = 𝑤 → (𝜑 → ∀𝑧(𝑧 = 𝑤𝜑)))
4 pm2.27 41 . . . 4 (𝑧 = 𝑤 → ((𝑧 = 𝑤𝜑) → 𝜑))
54al2imi 1733 . . 3 (∀𝑧 𝑧 = 𝑤 → (∀𝑧(𝑧 = 𝑤𝜑) → ∀𝑧𝜑))
63, 5syld 46 . 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 1473 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696 This theorem is referenced by:  axc16  2120  axc16gb  2121  axc16nf  2122  axc11v  2123  axc11rv  2124  aevOLD  2148  axc16nfALT  2311
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