Theorem bj-aecomsv 34990

Description: Version of aecoms 2428 with a disjoint variable condition, provable from Tarski's FOL. The corresponding version of naecoms 2429 should not be very useful since ¬ ∀𝑥𝑥 = 𝑦, DV (𝑥, 𝑦) is true when the universe has at least two objects (see dtru 5359). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-aecomsv.1	⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑)

Assertion

Ref	Expression
bj-aecomsv	⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-aecomsv

Step	Hyp	Ref	Expression
1		aevlem 2058	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑥 = 𝑦)
2		bj-aecomsv.1	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑)
3	1, 2	syl 17	1 ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑)

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 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783

This theorem is referenced by:  bj-axc11v  34991