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Theorem sbc2rexgOLD 42202
Description: Exchange a substitution with two 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
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 3489 . 2 (𝐴𝑉𝐴 ∈ V)
2 sbcrexgOLD 42199 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑎]𝑏𝐵𝑐𝐶 𝜑 ↔ ∃𝑏𝐵 [𝐴 / 𝑎]𝑐𝐶 𝜑))
3 sbcrexgOLD 42199 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑎]𝑐𝐶 𝜑 ↔ ∃𝑐𝐶 [𝐴 / 𝑎]𝜑))
43rexbidv 3174 . . 3 (𝐴 ∈ V → (∃𝑏𝐵 [𝐴 / 𝑎]𝑐𝐶 𝜑 ↔ ∃𝑏𝐵𝑐𝐶 [𝐴 / 𝑎]𝜑))
52, 4bitrd 279 . 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 205  wcel 2099  wrex 3066  Vcvv 3470  [wsbc 3775
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 2167  ax-13 2367  ax-ext 2699
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 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3058  df-rex 3067  df-v 3472  df-sbc 3776
This theorem is referenced by:  sbc4rexgOLD  42204
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