Theorem eceq2d 6740

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	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
eceq2d	⊢ (𝜑 → [𝐶]𝐴 = [𝐶]𝐵)

Proof of Theorem eceq2d

Step	Hyp	Ref	Expression
1		eceq2d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		eceq2 6738	. 2 ⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)
3	1, 2	syl 14	1 ⊢ (𝜑 → [𝐶]𝐴 = [𝐶]𝐵)

Syntax hints:   → wi 4   = wceq 1397  [cec 6699

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-ec 6703

This theorem is referenced by: (None)