Theorem axc11v 2298

Description: Version of axc11 2460 with a disjoint variable condition on 𝑥 and 𝑦, which is provable, on top of { ax-1 6-- ax-7 2027 }, from ax12v 2212 (contrary to axc11 2460 which seems to require the full ax-12 2211 and ax-13 2402). (Contributed by NM, 16-May-2008.) (Revised by BJ, 6-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.)

Assertion

Ref	Expression
axc11v	⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem axc11v

Step	Hyp	Ref	Expression
1		axc16g 2294	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑦𝜑))
2	1	spsd 2221	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-12 2211

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799

This theorem is referenced by:  dral1v  2399