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Theorem axc11rv 2177
 Description: Version of axc11r 2223 with a disjoint variable condition on 𝑥 and 𝑦, which is provable, on top of { ax-1 6-- ax-7 1981 }, from ax12v 2088 (contrary to axc11 2347 which seems to require the full ax-12 2087). (Contributed by BJ, 6-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.)
Assertion
Ref Expression
axc11rv (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem axc11rv
StepHypRef Expression
1 axc16g 2172 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
21spsd 2095 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1521 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745 This theorem is referenced by:  axc11vOLD  2179
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