Theorem equcomi 2123

Description: Commutative law for equality. Equality is a symmetric relation. Lemma 3 of [KalishMontague] p. 85. See also Lemma 7 of [Tarski] p. 69. (Contributed by NM, 10-Jan-1993.) (Revised by NM, 9-Apr-2017.)

Assertion

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

Proof of Theorem equcomi

Step	Hyp	Ref	Expression
1		equid 2118	. 2 ⊢ 𝑥 = 𝑥
2		ax7 2122	. 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 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1881

This theorem is referenced by:  equcom  2124  equcoms  2126  ax13dgen2  2191  cbv2h  2423  axc16i  2458  equsb2  2501  axsep  5005  rext  5138  soxp  7555  axextnd  9729  prodmo  15040  mpt2matmul  20621  finminlem  32852  bj-ssbid2ALT  33183  axc11n11  33208  axc11n11r  33209  bj-cbv2hv  33265  bj-axsep  33319  ax6er  33345  poimirlem25  33979  axc11nfromc11  35002  aev-o  35007