Theorem csbeq12dv 3842

Description: Formula-building inference for class substitution. (Contributed by SN, 3-Nov-2023.)

Hypotheses

Ref	Expression
csbeq12dv.1	⊢ (𝜑 → 𝐴 = 𝐶)
csbeq12dv.2	⊢ (𝜑 → 𝐵 = 𝐷)

Assertion

Ref	Expression
csbeq12dv	⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐷)

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem csbeq12dv

Step	Hyp	Ref	Expression
1		csbeq12dv.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐶)
2	1	csbeq1d 3837	. 2 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐵)
3		csbeq12dv.2	. . 3 ⊢ (𝜑 → 𝐵 = 𝐷)
4	3	csbeq2dv 3840	. 2 ⊢ (𝜑 → ⦋𝐶 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐷)
5	2, 4	eqtrd 2776	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐷)

Syntax hints:   → wi 4   = wceq 1548  ⦋csb 3833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-sbc 3726  df-csb 3834

This theorem is referenced by:  bpolylem  16008  selvffval  22098  selvfval  22099  selvval  22100  cbvitgv  25766  mulsval  28123  precsexlemcbv  28220  precsexlem3  28223  ttgval  28965  itgeq12sdv  36462  cbvitgvw2  36491  cbvitgdavw  36524  cbvitgdavw2  36540  poimirlem16  38018  poimirlem17  38019  poimirlem19  38021  poimirlem20  38022  isprimroot  42593  fmpocos  42735  grtri  48445  dfswapf2  49765  dfinito4  50005