Theorem bj-axc11v 33251

Description: Version of axc11 2437 with a disjoint variable condition, which does not require ax-13 2377 nor ax-10 2185. Remark: the following theorems (hbae 2438, nfae 2439, hbnae 2440, nfnae 2441, hbnaes 2442) would need to be totally unbundled to be proved without ax-13 2377, hence would be simple consequences of ax-5 2006 or nfv 2010. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-axc11v

Step	Hyp	Ref	Expression
1		axc11r 2376	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	bj-aecomsv 33250	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:  bj-dral1v  33252