Theorem eqtr4d 2388

Description: An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)

Hypotheses

Ref	Expression
eqtr4d.1	⊢ (φ → A = B)
eqtr4d.2	⊢ (φ → C = B)

Assertion

Ref	Expression
eqtr4d	⊢ (φ → A = C)

Proof of Theorem eqtr4d

Step	Hyp	Ref	Expression
1		eqtr4d.1	. 2 ⊢ (φ → A = B)
2		eqtr4d.2	. . 3 ⊢ (φ → C = B)
3	2	eqcomd 2358	. 2 ⊢ (φ → B = C)
4	1, 3	eqtrd 2385	1 ⊢ (φ → A = C)

Syntax hints:   → wi 4   = wceq 1642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-cleq 2346

This theorem is referenced by:  3eqtr2d  2391  3eqtr2rd  2392  3eqtr4d  2395  3eqtr4rd  2396  3eqtr4a  2411  sbnfc2  3196  ifsb  3671  ifeq1da  3687  ifeq2da  3688  ifnot  3700  ifan  3701  ifor  3702  setswith  4321  iotauni  4351  dfiota3  4370  dfiota4  4372  nnsucelrlem3  4426  nnpw1ex  4484  tfin11  4493  sfin112  4529  fveqres  5355  funfv  5375  fsn2  5434  fvunsn  5444  fconst2g  5452  ndmovcom  5617  ndmovass  5618  ndmovdistr  5619  fvmpti  5699  fvmptex  5721  fvfullfun  5864  uniqs2  5985  enprmaplem3  6078  nchoicelem7  6295  nchoicelem15  6303  nchoicelem17  6305  fnfrec  6320