Theorem equequ2 1759

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 1756	. 2 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → 𝑧 = 𝑦))
2		equtrr 1756	. . 3 ⊢ (𝑦 = 𝑥 → (𝑧 = 𝑦 → 𝑧 = 𝑥))
3	2	equcoms 1754	. 2 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑦 → 𝑧 = 𝑥))
4	1, 3	impbid 129	1 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 ↔ 𝑧 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1495  ax-ie2 1540  ax-8 1550  ax-17 1572  ax-i9 1576

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  ax11v2  1866  ax11v  1873  ax11ev  1874  equs5or  1876  eujust  2079  euf  2082  mo23  2119  eleq1w  2290  cbvabw  2352  csbcow  3135  disjiun  4077  iotaval  5289  dffun4f  5333  dff13f  5893  supmoti  7156  isoti  7170  nninfwlpoim  7342  exmidontriim  7403  netap  7436  ennnfonelemr  12989  ctinf  12996  infpn2  13022  lgseisenlem2  15744