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Theorem 2sbcrex 42082
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 3864 . . 3 ([𝐵 / 𝑏]𝑐𝐶 𝜑 ↔ ∃𝑐𝐶 [𝐵 / 𝑏]𝜑)
21sbcbii 3832 . 2 ([𝐴 / 𝑎][𝐵 / 𝑏]𝑐𝐶 𝜑[𝐴 / 𝑎]𝑐𝐶 [𝐵 / 𝑏]𝜑)
3 sbcrex 3864 . 2 ([𝐴 / 𝑎]𝑐𝐶 [𝐵 / 𝑏]𝜑 ↔ ∃𝑐𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑)
42, 3bitri 275 1 ([𝐴 / 𝑎][𝐵 / 𝑏]𝑐𝐶 𝜑 ↔ ∃𝑐𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wrex 3064  [wsbc 3772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-v 3470  df-sbc 3773
This theorem is referenced by:  2rexfrabdioph  42094  4rexfrabdioph  42096
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