Theorem axc11v 2259

Description: Version of axc11 2431 with a disjoint variable condition on 𝑥 and 𝑦, which is provable, on top of { ax-1 6-- ax-7 2014 }, from ax12v 2175 (contrary to axc11 2431 which seems to require the full ax-12 2174 and ax-13 2373). (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 2255	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑦𝜑))
2	1	spsd 2183	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-12 2174

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1786

This theorem is referenced by:  dral1v  2368  dral1vOLD  2369