Theorem eceq2d 6631

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 6629	. 2 |- ( A = B -> [ C ] A = [ C ] B )
3	1, 2	syl 14	1 |- ( ph -> [ C ] A = [ C ] B )

Syntax hints:   -> wi 4   = wceq 1364   [cec 6590

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-cnv 4671  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-ec 6594

This theorem is referenced by: (None)