Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-sbceqgALT Structured version   Visualization version   GIF version

Theorem bj-sbceqgALT 34223
 Description: Distribute proper substitution through an equality relation. Alternate proof of sbceqg 4364. (Contributed by BJ, 6-Oct-2018.) Proof modification is discouraged to avoid using sbceqg 4364, but the Metamath program "MM-PA> MINIMIZE_WITH * / EXCEPT sbceqg" command is ok. (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-sbceqgALT (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))

Proof of Theorem bj-sbceqgALT
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2818 . . . . . 6 (𝐵 = 𝐶 ↔ ∀𝑦(𝑦𝐵𝑦𝐶))
21sbcth 3790 . . . . 5 (𝐴𝑉[𝐴 / 𝑥](𝐵 = 𝐶 ↔ ∀𝑦(𝑦𝐵𝑦𝐶)))
3 sbcbig 3826 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥](𝐵 = 𝐶 ↔ ∀𝑦(𝑦𝐵𝑦𝐶)) ↔ ([𝐴 / 𝑥]𝐵 = 𝐶[𝐴 / 𝑥]𝑦(𝑦𝐵𝑦𝐶))))
42, 3mpbid 234 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶[𝐴 / 𝑥]𝑦(𝑦𝐵𝑦𝐶)))
5 sbcal 3836 . . . 4 ([𝐴 / 𝑥]𝑦(𝑦𝐵𝑦𝐶) ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝐵𝑦𝐶))
64, 5syl6bb 289 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝐵𝑦𝐶)))
7 sbcbig 3826 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥](𝑦𝐵𝑦𝐶) ↔ ([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶)))
87albidv 1920 . . 3 (𝐴𝑉 → (∀𝑦[𝐴 / 𝑥](𝑦𝐵𝑦𝐶) ↔ ∀𝑦([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶)))
9 sbcel2 4370 . . . . . 6 ([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵)
109a1i 11 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵))
11 sbcel2 4370 . . . . . 6 ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶)
1211a1i 11 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶))
1310, 12bibi12d 348 . . . 4 (𝐴𝑉 → (([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶) ↔ (𝑦𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))
1413albidv 1920 . . 3 (𝐴𝑉 → (∀𝑦([𝐴 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐶) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))
156, 8, 143bitrd 307 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))
16 dfcleq 2818 . 2 (𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))
1715, 16syl6bbr 291 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1534   = wceq 1536   ∈ wcel 2113  [wsbc 3775  ⦋csb 3886 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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-nul 4295 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator