Theorem csbeq12dv 3854

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 3849	. 2 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐵)
3		csbeq12dv.2	. . 3 ⊢ (𝜑 → 𝐵 = 𝐷)
4	3	csbeq2dv 3852	. 2 ⊢ (𝜑 → ⦋𝐶 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐷)
5	2, 4	eqtrd 2766	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐷)

Syntax hints:   → wi 4   = wceq 1541  ⦋csb 3845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-sbc 3737  df-csb 3846

This theorem is referenced by:  bpolylem  15955  selvffval  22048  selvfval  22049  selvval  22050  cbvitgv  25705  mulsval  28048  precsexlemcbv  28144  precsexlem3  28147  ttgval  28853  itgeq12sdv  36263  cbvitgvw2  36292  cbvitgdavw  36325  cbvitgdavw2  36341  poimirlem16  37675  poimirlem17  37676  poimirlem19  37678  poimirlem20  37679  isprimroot  42185  fmpocos  42326  grtri  48039  dfswapf2  49361  dfinito4  49601