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Theorem sbc4rexgOLD 39265
Description: Exchange a substitution with four existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 7187 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 3510 . 2 (𝐴𝑉𝐴 ∈ V)
2 sbc2rexgOLD 39263 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑎]𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 𝜑 ↔ ∃𝑏𝐵𝑐𝐶 [𝐴 / 𝑎]𝑑𝐷𝑒𝐸 𝜑))
3 sbc2rexgOLD 39263 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑎]𝑑𝐷𝑒𝐸 𝜑 ↔ ∃𝑑𝐷𝑒𝐸 [𝐴 / 𝑎]𝜑))
432rexbidv 3297 . . 3 (𝐴 ∈ V → (∃𝑏𝐵𝑐𝐶 [𝐴 / 𝑎]𝑑𝐷𝑒𝐸 𝜑 ↔ ∃𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸 [𝐴 / 𝑎]𝜑))
52, 4bitrd 280 . 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 207  wcel 2105  wrex 3136  Vcvv 3492  [wsbc 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-v 3494  df-sbc 3770
This theorem is referenced by: (None)
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