Theorem csbeq12dv 3883

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 3878	. 2 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐵)
3		csbeq12dv.2	. . 3 ⊢ (𝜑 → 𝐵 = 𝐷)
4	3	csbeq2dv 3881	. 2 ⊢ (𝜑 → ⦋𝐶 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐷)
5	2, 4	eqtrd 2770	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐷)

Syntax hints:   → wi 4   = wceq 1540  ⦋csb 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-sbc 3766  df-csb 3875

This theorem is referenced by:  bpolylem  16064  selvffval  22071  selvfval  22072  selvval  22073  cbvitgv  25730  mulsval  28064  precsexlemcbv  28160  precsexlem3  28163  ttgval  28854  itgeq12sdv  36237  cbvitgvw2  36266  cbvitgdavw  36299  cbvitgdavw2  36315  poimirlem16  37660  poimirlem17  37661  poimirlem19  37663  poimirlem20  37664  isprimroot  42106  fmpocos  42285  grtri  47952  dfswapf2  49178  dfinito4  49386