Theorem bj-axc11v 34970

Description: Version of axc11 2431 with a disjoint variable condition, which does not require ax-13 2373 nor ax-10 2140. Remark: the following theorems (hbae 2432, nfae 2434, hbnae 2433, nfnae 2435, hbnaes 2436) would need to be totally unbundled to be proved without ax-13 2373, hence would be simple consequences of ax-5 1916 or nfv 1920. (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 2260	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	bj-aecomsv 34969	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: (None)