Theorem bj-aecomsv 34132

Description: Version of aecoms 2450 with a disjoint variable condition, provable from Tarski's FOL. The corresponding version of naecoms 2451 should not be very useful since ¬ ∀𝑥𝑥 = 𝑦, DV (𝑥, 𝑦) is true when the universe has at least two objects (see dtru 5273). (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 2060	. 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 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781

This theorem is referenced by:  bj-axc11v  34133