Theorem csbeq12dv 3846

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 3841	. 2 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐵)
3		csbeq12dv.2	. . 3 ⊢ (𝜑 → 𝐵 = 𝐷)
4	3	csbeq2dv 3844	. 2 ⊢ (𝜑 → ⦋𝐶 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐷)
5	2, 4	eqtrd 2771	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐷)

Syntax hints:   → wi 4   = wceq 1542  ⦋csb 3837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-sbc 3729  df-csb 3838

This theorem is referenced by:  bpolylem  16013  selvffval  22099  selvfval  22100  selvval  22101  cbvitgv  25744  mulsval  28101  precsexlemcbv  28198  precsexlem3  28201  ttgval  28943  itgeq12sdv  36401  cbvitgvw2  36430  cbvitgdavw  36463  cbvitgdavw2  36479  poimirlem16  37957  poimirlem17  37958  poimirlem19  37960  poimirlem20  37961  isprimroot  42532  fmpocos  42675  grtri  48416  dfswapf2  49736  dfinito4  49976