Theorem equequ2 1701

Description: An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
equequ2	⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 ↔ 𝑧 = 𝑦))

Proof of Theorem equequ2

Step	Hyp	Ref	Expression
1		equtrr 1698	. 2 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → 𝑧 = 𝑦))
2		equtrr 1698	. . 3 ⊢ (𝑦 = 𝑥 → (𝑧 = 𝑦 → 𝑧 = 𝑥))
3	2	equcoms 1696	. 2 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑦 → 𝑧 = 𝑥))
4	1, 3	impbid 128	1 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 ↔ 𝑧 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1437  ax-ie2 1482  ax-8 1492  ax-17 1514  ax-i9 1518

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  ax11v2  1808  ax11v  1815  ax11ev  1816  equs5or  1818  eujust  2016  euf  2019  mo23  2055  eleq1w  2227  cbvabw  2289  csbcow  3056  disjiun  3977  iotaval  5164  dffun4f  5204  dff13f  5738  supmoti  6958  isoti  6972  exmidontriim  7181  ennnfonelemr  12356  ctinf  12363  infpn2  12389