Theorem equcomd 2054

Description: Deduction form of equcom 2053, symmetry of equality. For the versions for classes, see eqcom 2770 and eqcomd 2769. (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 2053	. 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 1945  ax-6 2002  ax-7 2043

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  sndisj  5098  fsumcom2  15972  fprodcom2  16183  catideu  17881  pospo  18549  dprdfcntz  20264  ordtt1  23730  eengtrkg  29597  cusgrfilem2  30070  frgr2wwlk1  30963  ssmxidl  34083  gonar  36340  elpssmo  36649  bj-nfcsym  38002  exidu1  38993  rngoideu  39040  2reu8i  48399  ichnreuop  48769  sprsymrelf1lem  48788  oppcthinendcALT  50791