Theorem eceq2d 6672

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 6670	. 2 |- ( A = B -> [ C ] A = [ C ] B )
3	1, 2	syl 14	1 |- ( ph -> [ C ] A = [ C ] B )

Syntax hints:   -> wi 4   = wceq 1373   [cec 6631

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-cnv 4691  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-ec 6635

This theorem is referenced by: (None)