Theorem axc11rv 2303

Description: Version of axc11r 2402 with a disjoint variable condition on 𝑥 and 𝑦, which is provable, on top of { ax-1 6-- ax-7 2031 }, from ax12v 2216 (contrary to axc11 2464 which seems to require the full ax-12 2215 and ax-13 2406, and to axc11r 2402 which seems to require the full ax-12 2215). (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

Step	Hyp	Ref	Expression
1		axc16 2299	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
2	1	spsd 2225	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-12 2215

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803

This theorem is referenced by:  dral1v  2403  bj-axc11v  37306