Theorem equcomd 2051

Description: Deduction form of equcom 2050, symmetry of equality. For the versions for classes, see eqcom 2767 and eqcomd 2766. (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 2050	. 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 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812

This theorem is used by:  sndisj  5094  fsumcom2  15946  fprodcom2  16157  catideu  17855  pospo  18523  dprdfcntz  20237  ordtt1  23703  eengtrkg  29570  cusgrfilem2  30043  frgr2wwlk1  30936  ssmxidl  34046  gonar  36247  bj-nfcsym  37904  exidu1  38884  rngoideu  38931  2reu8i  48277  ichnreuop  48648  sprsymrelf1lem  48667  oppcthinendcALT  50670