Theorem axc11v 2286

Description: Version of axc11 2437 with a disjoint variable condition on 𝑥 and 𝑦, which is provable, on top of { ax-1 6-- ax-7 2107 }, from ax12v 2214 (contrary to axc11 2437 which seems to require the full ax-12 2213 and ax-13 2377). (Contributed 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 2282	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑦𝜑))
2	1	spsd 2221	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-12 2213

This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876

This theorem is referenced by: (None)