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Theorem 2sbcrex 38193
Description: Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.)
Assertion
Ref Expression
2sbcrex ([𝐴 / 𝑎][𝐵 / 𝑏]𝑐𝐶 𝜑 ↔ ∃𝑐𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑)
Distinct variable groups:   𝐴,𝑐   𝐵,𝑐   𝐶,𝑏   𝑎,𝑐   𝑏,𝑐   𝐶,𝑎
Allowed substitution hints:   𝜑(𝑎,𝑏,𝑐)   𝐴(𝑎,𝑏)   𝐵(𝑎,𝑏)   𝐶(𝑐)

Proof of Theorem 2sbcrex
StepHypRef Expression
1 sbcrex 3739 . . 3 ([𝐵 / 𝑏]𝑐𝐶 𝜑 ↔ ∃𝑐𝐶 [𝐵 / 𝑏]𝜑)
21sbcbii 3719 . 2 ([𝐴 / 𝑎][𝐵 / 𝑏]𝑐𝐶 𝜑[𝐴 / 𝑎]𝑐𝐶 [𝐵 / 𝑏]𝜑)
3 sbcrex 3739 . 2 ([𝐴 / 𝑎]𝑐𝐶 [𝐵 / 𝑏]𝜑 ↔ ∃𝑐𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑)
42, 3bitri 267 1 ([𝐴 / 𝑎][𝐵 / 𝑏]𝑐𝐶 𝜑 ↔ ∃𝑐𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wrex 3119  [wsbc 3663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-v 3417  df-sbc 3664
This theorem is referenced by:  2rexfrabdioph  38205  4rexfrabdioph  38207
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