Theorem eqtr3di 2277

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 2233	. 2 ⊢ 𝐶 = 𝐴
3		eqtr3di.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
4	2, 3	eqtr2id 2275	1 ⊢ (𝜑 → 𝐵 = 𝐶)

Syntax hints:   → wi 4   = wceq 1395

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 1493  ax-gen 1495  ax-4 1556  ax-17 1572  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222

This theorem is referenced by:  bm2.5ii  4588  resdmdfsn  5048  f0dom0  5521  f1o00  5610  fmpt  5787  fmptsn  5832  resfunexg  5864  mapsn  6845  sbthlemi4  7138  sbthlemi6  7140  pm54.43  7374  prarloclem5  7698  recexprlem1ssl  7831  recexprlem1ssu  7832  iooval2  10123  hashsng  11032  zfz1isolem1  11075  resqrexlemover  11536  isumclim3  11949  algrp1  12583  pythagtriplem1  12803  ressbasid  13118  ressval3d  13120  ressressg  13123  tangtx  15527  coskpi  15537  lgsquadlem2  15772  2omap  16418  pw1map  16420  subctctexmid  16425