Theorem eceq2d 8515

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 8513	. 2 ⊢ (𝐴 = 𝐵 → [𝐶]𝐴 = [𝐶]𝐵)
3	1, 2	syl 17	1 ⊢ (𝜑 → [𝐶]𝐴 = [𝐶]𝐵)

Syntax hints:   → wi 4   = wceq 1542  [cec 8471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ec 8475

This theorem is referenced by:  vrgpfval  19362  quslsm  31581  releldmqscoss  36760  prjspeclsp  40440