Theorem bj-axc11v 37691

Description: Version of axc11 2460 with a disjoint variable condition, which does not require ax-13 2402 nor ax-10 2178. Remark: the following theorems (hbae 2461, nfae 2463, hbnae 2462, nfnae 2464, hbnaes 2465) would need to be totally unbundled to be proved without ax-13 2402, hence would be simple consequences of ax-5 1943 or nfv 1947. (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 2300	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
2	1	bj-aecomsv 37690	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2213

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by: (None)