Theorem axc11rv 2262

Description: Version of axc11r 2368 with a disjoint variable condition on 𝑥 and 𝑦, which is provable, on top of { ax-1 6-- ax-7 2004 }, from ax12v 2175 (contrary to axc11 2432 which seems to require the full ax-12 2174 and ax-13 2374, and to axc11r 2368 which seems to require the full ax-12 2174). (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 2258	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
2	1	spsd 2184	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-12 2174

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776

This theorem is referenced by:  dral1v  2369  dral1vOLD  2370  bj-axc11v  36791