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Theorem sbc2rex 39374
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 3856 . 2 ([𝐴 / 𝑎]𝑏𝐵𝑐𝐶 𝜑 ↔ ∃𝑏𝐵 [𝐴 / 𝑎]𝑐𝐶 𝜑)
2 sbcrex 3856 . . 3 ([𝐴 / 𝑎]𝑐𝐶 𝜑 ↔ ∃𝑐𝐶 [𝐴 / 𝑎]𝜑)
32rexbii 3245 . 2 (∃𝑏𝐵 [𝐴 / 𝑎]𝑐𝐶 𝜑 ↔ ∃𝑏𝐵𝑐𝐶 [𝐴 / 𝑎]𝜑)
41, 3bitri 277 1 ([𝐴 / 𝑎]𝑏𝐵𝑐𝐶 𝜑 ↔ ∃𝑏𝐵𝑐𝐶 [𝐴 / 𝑎]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 208  wrex 3137  [wsbc 3770
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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-v 3495  df-sbc 3771
This theorem is referenced by:  sbc4rex  39376  3rexfrabdioph  39384  4rexfrabdioph  39385  7rexfrabdioph  39387
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