Theorem eqtr3di 2244

Description: An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)

Hypotheses

Ref	Expression
eqtr3di.1	⊢ (𝜑 → 𝐴 = 𝐵)
eqtr3di.2	⊢ 𝐴 = 𝐶

Assertion

Ref	Expression
eqtr3di	⊢ (𝜑 → 𝐵 = 𝐶)

Proof of Theorem eqtr3di

Step	Hyp	Ref	Expression
1		eqtr3di.2	. . 3 ⊢ 𝐴 = 𝐶
2	1	eqcomi 2200	. 2 ⊢ 𝐶 = 𝐴
3		eqtr3di.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
4	2, 3	eqtr2id 2242	1 ⊢ (𝜑 → 𝐵 = 𝐶)

Syntax hints:   → wi 4   = wceq 1364

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-5 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189

This theorem is referenced by:  bm2.5ii  4533  resdmdfsn  4990  f0dom0  5454  f1o00  5542  fmpt  5715  fmptsn  5754  resfunexg  5786  mapsn  6758  sbthlemi4  7035  sbthlemi6  7037  pm54.43  7269  prarloclem5  7584  recexprlem1ssl  7717  recexprlem1ssu  7718  iooval2  10007  hashsng  10907  zfz1isolem1  10949  resqrexlemover  11192  isumclim3  11605  algrp1  12239  pythagtriplem1  12459  ressbasid  12773  ressval3d  12775  ressressg  12778  tangtx  15158  coskpi  15168  lgsquadlem2  15403  2omap  15726  subctctexmid  15731