Theorem equequ1 1758

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

Assertion

Ref	Expression
equequ1	⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))

Proof of Theorem equequ1

Step	Hyp	Ref	Expression
1		ax-8 1550	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
2		equtr 1755	. 2 ⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝑥 = 𝑧))
3	1, 2	impbid 129	1 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))

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:  equveli  1805  drsb1  1845  equsb3lem  2001  euequ1  2173  axext3  2212  cbvreuvw  2771  reu6  2992  reu7  2998  reu8nf  3110  disjiun  4078  cbviota  5286  dff13f  5903  poxp  6389  dcdifsnid  6663  supmoti  7176  isoti  7190  nninfwlpoim  7362  exmidontriimlem3  7421  exmidontriim  7423  netap  7456  fsum2dlemstep  11966  ennnfonelemr  13015  ctinf  13022  reap0  16540