Users' Mathboxes Mathbox for metakunt < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rspcsbnea Structured version   Visualization version   GIF version

Theorem rspcsbnea 42113
Description: Special case related to rspsbc 3888. (Contributed by metakunt, 5-May-2025.)
Assertion
Ref Expression
rspcsbnea ((𝐴𝐵 ∧ ∀𝑥𝐵 𝐶𝐷) → 𝐴 / 𝑥𝐶𝐷)
Distinct variable groups:   𝑥,𝐵   𝑥,𝐷
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem rspcsbnea
StepHypRef Expression
1 rspsbc 3888 . . 3 (𝐴𝐵 → (∀𝑥𝐵 𝐶𝐷[𝐴 / 𝑥]𝐶𝐷))
2 df-ne 2939 . . . . . . 7 (𝐶𝐷 ↔ ¬ 𝐶 = 𝐷)
32sbcbii 3852 . . . . . 6 ([𝐴 / 𝑥]𝐶𝐷[𝐴 / 𝑥] ¬ 𝐶 = 𝐷)
43a1i 11 . . . . 5 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷[𝐴 / 𝑥] ¬ 𝐶 = 𝐷))
5 sbcng 3842 . . . . . 6 (𝐴𝐵 → ([𝐴 / 𝑥] ¬ 𝐶 = 𝐷 ↔ ¬ [𝐴 / 𝑥]𝐶 = 𝐷))
6 sbceq1g 4423 . . . . . . 7 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶 = 𝐷𝐴 / 𝑥𝐶 = 𝐷))
76notbid 318 . . . . . 6 (𝐴𝐵 → (¬ [𝐴 / 𝑥]𝐶 = 𝐷 ↔ ¬ 𝐴 / 𝑥𝐶 = 𝐷))
85, 7bitrd 279 . . . . 5 (𝐴𝐵 → ([𝐴 / 𝑥] ¬ 𝐶 = 𝐷 ↔ ¬ 𝐴 / 𝑥𝐶 = 𝐷))
94, 8bitrd 279 . . . 4 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷 ↔ ¬ 𝐴 / 𝑥𝐶 = 𝐷))
10 biidd 262 . . . . 5 (𝐴𝐵 → (𝐴 / 𝑥𝐶 = 𝐷𝐴 / 𝑥𝐶 = 𝐷))
1110necon3bbid 2976 . . . 4 (𝐴𝐵 → (¬ 𝐴 / 𝑥𝐶 = 𝐷𝐴 / 𝑥𝐶𝐷))
129, 11bitrd 279 . . 3 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐷))
131, 12sylibd 239 . 2 (𝐴𝐵 → (∀𝑥𝐵 𝐶𝐷𝐴 / 𝑥𝐶𝐷))
1413imp 406 1 ((𝐴𝐵 ∧ ∀𝑥𝐵 𝐶𝐷) → 𝐴 / 𝑥𝐶𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  [wsbc 3791  csb 3908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-v 3480  df-sbc 3792  df-csb 3909
This theorem is referenced by:  idomnnzgmulnz  42115  deg1gprod  42122
  Copyright terms: Public domain W3C validator