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Theorem bj-axc16g16 32901
 Description: Proof of axc16g 2245 from { ax-1 6-- ax-7 2054, axc16 2246 }. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axc16g16 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem bj-axc16g16
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 aevlem 2096 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑡)
2 axc16 2246 . 2 (∀𝑧 𝑧 = 𝑡 → (𝜑 → ∀𝑧𝜑))
31, 2syl 17 1 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1594 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-12 2160 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1818 This theorem is referenced by: (None)
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