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

Theorem bj-sbeq 33321
Description: Distribute proper substitution through an equality relation. (See sbceqg 4145). (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-sbeq ([𝑦 / 𝑥]𝐴 = 𝐵𝑦 / 𝑥𝐴 = 𝑦 / 𝑥𝐵)

Proof of Theorem bj-sbeq
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2759 . . . . 5 (𝐴 = 𝐵 ↔ ∀𝑧(𝑧𝐴𝑧𝐵))
21sbbii 2069 . . . 4 ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ [𝑦 / 𝑥]∀𝑧(𝑧𝐴𝑧𝐵))
3 sbsbc 3600 . . . 4 ([𝑦 / 𝑥]∀𝑧(𝑧𝐴𝑧𝐵) ↔ [𝑦 / 𝑥]𝑧(𝑧𝐴𝑧𝐵))
4 sbcal 3646 . . . 4 ([𝑦 / 𝑥]𝑧(𝑧𝐴𝑧𝐵) ↔ ∀𝑧[𝑦 / 𝑥](𝑧𝐴𝑧𝐵))
52, 3, 43bitri 288 . . 3 ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ ∀𝑧[𝑦 / 𝑥](𝑧𝐴𝑧𝐵))
6 vex 3353 . . . . 5 𝑦 ∈ V
7 sbcbig 3641 . . . . 5 (𝑦 ∈ V → ([𝑦 / 𝑥](𝑧𝐴𝑧𝐵) ↔ ([𝑦 / 𝑥]𝑧𝐴[𝑦 / 𝑥]𝑧𝐵)))
86, 7ax-mp 5 . . . 4 ([𝑦 / 𝑥](𝑧𝐴𝑧𝐵) ↔ ([𝑦 / 𝑥]𝑧𝐴[𝑦 / 𝑥]𝑧𝐵))
98albii 1914 . . 3 (∀𝑧[𝑦 / 𝑥](𝑧𝐴𝑧𝐵) ↔ ∀𝑧([𝑦 / 𝑥]𝑧𝐴[𝑦 / 𝑥]𝑧𝐵))
10 sbcel2 4150 . . . . 5 ([𝑦 / 𝑥]𝑧𝐴𝑧𝑦 / 𝑥𝐴)
11 sbcel2 4150 . . . . 5 ([𝑦 / 𝑥]𝑧𝐵𝑧𝑦 / 𝑥𝐵)
1210, 11bibi12i 330 . . . 4 (([𝑦 / 𝑥]𝑧𝐴[𝑦 / 𝑥]𝑧𝐵) ↔ (𝑧𝑦 / 𝑥𝐴𝑧𝑦 / 𝑥𝐵))
1312albii 1914 . . 3 (∀𝑧([𝑦 / 𝑥]𝑧𝐴[𝑦 / 𝑥]𝑧𝐵) ↔ ∀𝑧(𝑧𝑦 / 𝑥𝐴𝑧𝑦 / 𝑥𝐵))
145, 9, 133bitri 288 . 2 ([𝑦 / 𝑥]𝐴 = 𝐵 ↔ ∀𝑧(𝑧𝑦 / 𝑥𝐴𝑧𝑦 / 𝑥𝐵))
15 dfcleq 2759 . 2 (𝑦 / 𝑥𝐴 = 𝑦 / 𝑥𝐵 ↔ ∀𝑧(𝑧𝑦 / 𝑥𝐴𝑧𝑦 / 𝑥𝐵))
1614, 15bitr4i 269 1 ([𝑦 / 𝑥]𝐴 = 𝐵𝑦 / 𝑥𝐴 = 𝑦 / 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wal 1650   = wceq 1652  [wsb 2062  wcel 2155  Vcvv 3350  [wsbc 3596  csb 3691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-nul 4080
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator