Theorem equcomd 2052

Description: Deduction form of equcom 2051, symmetry of equality. For the versions for classes, see eqcom 2773 and eqcomd 2772. (Contributed by BJ, 6-Oct-2019.)

Hypothesis

Ref	Expression
equcomd.1	⊢ (𝜑 → 𝑥 = 𝑦)

Assertion

Ref	Expression
equcomd	⊢ (𝜑 → 𝑦 = 𝑥)

Proof of Theorem equcomd

Step	Hyp	Ref	Expression
1		equcomd.1	. 2 ⊢ (𝜑 → 𝑥 = 𝑦)
2		equcom 2051	. 2 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
3	1, 2	sylib 221	1 ⊢ (𝜑 → 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-an 402  df-ex 1813

This theorem is used by:  sndisj  5106  fsumcom2  15851  fprodcom2  16064  catideu  17756  pospo  18424  dprdfcntz  20118  ordtt1  23573  eengtrkg  29373  cusgrfilem2  29843  frgr2wwlk1  30717  ssmxidl  33788  gonar  35908  bj-nfcsym  37575  exidu1  38548  rngoideu  38595  2reu8i  47891  ichnreuop  48262  sprsymrelf1lem  48281  oppcthinendcALT  50260