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  6753  sbthlemi4  7030  sbthlemi6  7032  pm54.43  7262  prarloclem5  7572  recexprlem1ssl  7705  recexprlem1ssu  7706  iooval2  9995  hashsng  10895  zfz1isolem1  10937  resqrexlemover  11180  isumclim3  11593  algrp1  12227  pythagtriplem1  12447  ressbasid  12761  ressval3d  12763  ressressg  12766  tangtx  15121  coskpi  15131  lgsquadlem2  15366  2omap  15689  subctctexmid  15694