Theorem bj-axc11v 34135

Description: Version of axc11 2451 with a disjoint variable condition, which does not require ax-13 2389 nor ax-10 2144. Remark: the following theorems (hbae 2452, nfae 2454, hbnae 2453, nfnae 2455, hbnaes 2456) would need to be totally unbundled to be proved without ax-13 2389, hence would be simple consequences of ax-5 1910 or nfv 1914. (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 2265	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	bj-aecomsv 34134	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Syntax hints:   → wi 4  ∀wal 1534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780

This theorem is referenced by: (None)