Theorem bj-aecomsv 32746

Description: Version of aecoms 2312 with a dv condition, provable from Tarski's FOL. The corresponding version of naecoms 2313 should not be very useful since ¬ ∀𝑥𝑥 = 𝑦, DV(x, y) is true when the universe has at least two objects (see bj-dtru 32797). (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		bj-axc11nv 32745	. 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 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  bj-axc11v  32747