Theorem equcomiv 2047

Description: Weaker form of equcomi 2050 with a disjoint variable condition on 𝑥, 𝑦. This is an intermediate step and equcomi 2050 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 2045	. 2 ⊢ 𝑥 = 𝑥
2		ax7v2 2044	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑥 → 𝑦 = 𝑥))
3	1, 2	mpi 21	1 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)

This proof depends on syntax axioms:   → wi 4

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

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

This theorem is used by:  ax6evr  2048  pgnbgreunbgrlem1  48919