Theorem equcomiv 1941

Description: Weaker form of equcomi 1944 with a dv condition on 𝑥, 𝑦. This is an intermediate step and equcomi 1944 is fully recovered later. (Contributed by BJ, 7-Dec-2020.)

Assertion

Ref	Expression
equcomiv	⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)

Distinct variable group:   𝑥,𝑦

Proof of Theorem equcomiv

Step	Hyp	Ref	Expression
1		equid 1939	. 2 ⊢ 𝑥 = 𝑥
2		ax7v2 1938	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑥 → 𝑦 = 𝑥))
3	1, 2	mpi 20	1 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)

Syntax hints:   → wi 4

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-ex 1705

This theorem is referenced by:  ax6evr  1942