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Theorem sbc2rexgOLD 40124
 Description: Exchange a substitution with two existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 7192 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbc2rexgOLD (𝐴𝑉 → ([𝐴 / 𝑎]𝑏𝐵𝑐𝐶 𝜑 ↔ ∃𝑏𝐵𝑐𝐶 [𝐴 / 𝑎]𝜑))
Distinct variable groups:   𝐴,𝑏   𝐴,𝑐   𝐵,𝑎   𝐶,𝑎   𝑎,𝑏   𝑎,𝑐
Allowed substitution hints:   𝜑(𝑎,𝑏,𝑐)   𝐴(𝑎)   𝐵(𝑏,𝑐)   𝐶(𝑏,𝑐)   𝑉(𝑎,𝑏,𝑐)

Proof of Theorem sbc2rexgOLD
StepHypRef Expression
1 elex 3428 . 2 (𝐴𝑉𝐴 ∈ V)
2 sbcrexgOLD 40121 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑎]𝑏𝐵𝑐𝐶 𝜑 ↔ ∃𝑏𝐵 [𝐴 / 𝑎]𝑐𝐶 𝜑))
3 sbcrexgOLD 40121 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑎]𝑐𝐶 𝜑 ↔ ∃𝑐𝐶 [𝐴 / 𝑎]𝜑))
43rexbidv 3221 . . 3 (𝐴 ∈ V → (∃𝑏𝐵 [𝐴 / 𝑎]𝑐𝐶 𝜑 ↔ ∃𝑏𝐵𝑐𝐶 [𝐴 / 𝑎]𝜑))
52, 4bitrd 282 . 2 (𝐴 ∈ V → ([𝐴 / 𝑎]𝑏𝐵𝑐𝐶 𝜑 ↔ ∃𝑏𝐵𝑐𝐶 [𝐴 / 𝑎]𝜑))
61, 5syl 17 1 (𝐴𝑉 → ([𝐴 / 𝑎]𝑏𝐵𝑐𝐶 𝜑 ↔ ∃𝑏𝐵𝑐𝐶 [𝐴 / 𝑎]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2111  ∃wrex 3071  Vcvv 3409  [wsbc 3696 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3697 This theorem is referenced by:  sbc4rexgOLD  40126
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