Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sbc4rexgOLD Structured version   Visualization version   GIF version

Theorem sbc4rexgOLD 42182
Description: Exchange a substitution with four existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 7456 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbc4rexgOLD (𝐴𝑉 → ([𝐴 / 𝑎]𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 𝜑 ↔ ∃𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 [𝐴 / 𝑎]𝜑))
Distinct variable groups:   𝐴,𝑏   𝐴,𝑐   𝐵,𝑎   𝐶,𝑎   𝑎,𝑏   𝑎,𝑐   𝐴,𝑑   𝐴,𝑒   𝐷,𝑎   𝐸,𝑎   𝑎,𝑑   𝑒,𝑎
Allowed substitution hints:   𝜑(𝑒,𝑎,𝑏,𝑐,𝑑)   𝐴(𝑎)   𝐵(𝑒,𝑏,𝑐,𝑑)   𝐶(𝑒,𝑏,𝑐,𝑑)   𝐷(𝑒,𝑏,𝑐,𝑑)   𝐸(𝑒,𝑏,𝑐,𝑑)   𝑉(𝑒,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem sbc4rexgOLD
StepHypRef Expression
1 elex 3488 . 2 (𝐴𝑉𝐴 ∈ V)
2 sbc2rexgOLD 42180 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑎]𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 𝜑 ↔ ∃𝑏𝐵𝑐𝐶 [𝐴 / 𝑎]𝑑𝐷𝑒𝐸 𝜑))
3 sbc2rexgOLD 42180 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑎]𝑑𝐷𝑒𝐸 𝜑 ↔ ∃𝑑𝐷𝑒𝐸 [𝐴 / 𝑎]𝜑))
432rexbidv 3214 . . 3 (𝐴 ∈ V → (∃𝑏𝐵𝑐𝐶 [𝐴 / 𝑎]𝑑𝐷𝑒𝐸 𝜑 ↔ ∃𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 [𝐴 / 𝑎]𝜑))
52, 4bitrd 279 . 2 (𝐴 ∈ V → ([𝐴 / 𝑎]𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 𝜑 ↔ ∃𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 [𝐴 / 𝑎]𝜑))
61, 5syl 17 1 (𝐴𝑉 → ([𝐴 / 𝑎]𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 𝜑 ↔ ∃𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 [𝐴 / 𝑎]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2099  wrex 3065  Vcvv 3469  [wsbc 3774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-13 2366  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3057  df-rex 3066  df-v 3471  df-sbc 3775
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator