Theorem equequ2 1761

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 1758	. 2 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → 𝑧 = 𝑦))
2		equtrr 1758	. . 3 ⊢ (𝑦 = 𝑥 → (𝑧 = 𝑦 → 𝑧 = 𝑥))
3	2	equcoms 1756	. 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 1498  ax-ie2 1543  ax-8 1553  ax-17 1575  ax-i9 1579

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  ax11v2  1868  ax11v  1875  ax11ev  1876  equs5or  1878  eujust  2081  euf  2084  mo23  2121  eleq1w  2292  cbvabw  2355  csbcow  3139  disjiun  4088  iotaval  5305  dffun4f  5349  dff13f  5921  modom  7037  supmoti  7235  isoti  7249  nninfwlpoim  7421  exmidontriim  7483  netap  7516  ennnfonelemr  13107  ctinf  13114  infpn2  13140  lgseisenlem2  15873