Theorem bj-axc11v 36811

Description: Version of axc11 2434 with a disjoint variable condition, which does not require ax-13 2376 nor ax-10 2140. Remark: the following theorems (hbae 2435, nfae 2437, hbnae 2436, nfnae 2438, hbnaes 2439) would need to be totally unbundled to be proved without ax-13 2376, hence would be simple consequences of ax-5 1909 or nfv 1913. (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		axc11rv 2264	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	bj-aecomsv 36810	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-12 2176

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779

This theorem is referenced by: (None)