Theorem eceq2d 6658

Description: Equality theorem for the A-coset and B-coset of C, deduction version. (Contributed by Peter Mazsa, 23-Apr-2021.)

Hypothesis

Ref	Expression
eceq2d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
eceq2d	|- ( ph -> [ C ] A = [ C ] B )

Proof of Theorem eceq2d

Step	Hyp	Ref	Expression
1		eceq2d.1	. 2 |- ( ph -> A = B )
2		eceq2 6656	. 2 |- ( A = B -> [ C ] A = [ C ] B )
3	1, 2	syl 14	1 |- ( ph -> [ C ] A = [ C ] B )

Syntax hints:   -> wi 4   = wceq 1372   [cec 6617

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-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-cnv 4682  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-ec 6621

This theorem is referenced by: (None)