Theorem equequ1 1712

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 1504	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
2		equtr 1709	. 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 1449  ax-ie2 1494  ax-8 1504  ax-17 1526  ax-i9 1530

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  equveli  1759  drsb1  1799  equsb3lem  1950  euequ1  2121  axext3  2160  cbvreuvw  2711  reu6  2928  reu7  2934  disjiun  4000  cbviota  5185  dff13f  5773  poxp  6235  dcdifsnid  6507  supmoti  6994  isoti  7008  nninfwlpoim  7178  exmidontriimlem3  7224  exmidontriim  7226  netap  7255  fsum2dlemstep  11444  ennnfonelemr  12426  ctinf  12433  reap0  14845