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Theorem sbc2rex 36369
 Description: Exchange a substitution with two existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.)
Assertion
Ref Expression
sbc2rex ([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)

Proof of Theorem sbc2rex
StepHypRef Expression
1 sbcrex 3481 . 2 ([𝐴 / 𝑎]𝑏𝐵𝑐𝐶 𝜑 ↔ ∃𝑏𝐵 [𝐴 / 𝑎]𝑐𝐶 𝜑)
2 sbcrex 3481 . . 3 ([𝐴 / 𝑎]𝑐𝐶 𝜑 ↔ ∃𝑐𝐶 [𝐴 / 𝑎]𝜑)
32rexbii 3023 . 2 (∃𝑏𝐵 [𝐴 / 𝑎]𝑐𝐶 𝜑 ↔ ∃𝑏𝐵𝑐𝐶 [𝐴 / 𝑎]𝜑)
41, 3bitri 263 1 ([𝐴 / 𝑎]𝑏𝐵𝑐𝐶 𝜑 ↔ ∃𝑏𝐵𝑐𝐶 [𝐴 / 𝑎]𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  ∃wrex 2897  [wsbc 3402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403 This theorem is referenced by:  sbc4rex  36371  3rexfrabdioph  36379  4rexfrabdioph  36380  7rexfrabdioph  36382
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