Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbeq12dv Structured version   Visualization version   GIF version

Theorem csbeq12dv 3864
 Description: Formula-building inference for class substitution. (Contributed by SN, 3-Nov-2023.)
Hypotheses
Ref Expression
csbeq12dv.1 (𝜑𝐴 = 𝐶)
csbeq12dv.2 (𝜑𝐵 = 𝐷)
Assertion
Ref Expression
csbeq12dv (𝜑𝐴 / 𝑥𝐵 = 𝐶 / 𝑥𝐷)
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem csbeq12dv
StepHypRef Expression
1 csbeq12dv.1 . . 3 (𝜑𝐴 = 𝐶)
21csbeq1d 3859 . 2 (𝜑𝐴 / 𝑥𝐵 = 𝐶 / 𝑥𝐵)
3 csbeq12dv.2 . . 3 (𝜑𝐵 = 𝐷)
43csbeq2dv 3862 . 2 (𝜑𝐶 / 𝑥𝐵 = 𝐶 / 𝑥𝐷)
52, 4eqtrd 2857 1 (𝜑𝐴 / 𝑥𝐵 = 𝐶 / 𝑥𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  ⦋csb 3855 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-12 2178  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-sbc 3748  df-csb 3856 This theorem is referenced by:  selvffval  20786  selvfval  20787  selvval  20788
 Copyright terms: Public domain W3C validator