Theorem csbeq12dv 3861

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 3856	. 2 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐵)
3		csbeq12dv.2	. . 3 ⊢ (𝜑 → 𝐵 = 𝐷)
4	3	csbeq2dv 3859	. 2 ⊢ (𝜑 → ⦋𝐶 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐷)
5	2, 4	eqtrd 2797	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐷)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ⦋csb 3852

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-sbc 3744  df-csb 3853

This theorem is used by:  bpolylem  16108  rhmval0  20564  selvffval  22280  selvfval  22281  selvval  22282  cbvitgv  25947  mulsval  28313  precsexlemcbv  28410  precsexlem3  28413  ttgval  29235  nmulprop  36690  itgeq12sdv  36759  cbvitgvw2  36788  cbvitgdavw  36821  cbvitgdavw2  36837  poimirlem16  38315  poimirlem17  38316  poimirlem19  38318  poimirlem20  38319  isprimroot  42888  fmpocos  43032  grtri  48733  dfswapf2  50067  dfinito4  50307