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Theorem 2sbcrex 36256
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 3385 . . 3 ([𝐵 / 𝑏]𝑐𝐶 𝜑 ↔ ∃𝑐𝐶 [𝐵 / 𝑏]𝜑)
21sbcbii 3362 . 2 ([𝐴 / 𝑎][𝐵 / 𝑏]𝑐𝐶 𝜑[𝐴 / 𝑎]𝑐𝐶 [𝐵 / 𝑏]𝜑)
3 sbcrex 3385 . 2 ([𝐴 / 𝑎]𝑐𝐶 [𝐵 / 𝑏]𝜑 ↔ ∃𝑐𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑)
42, 3bitri 262 1 ([𝐴 / 𝑎][𝐵 / 𝑏]𝑐𝐶 𝜑 ↔ ∃𝑐𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wrex 2801  [wsbc 3306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-v 3079  df-sbc 3307
This theorem is referenced by:  2rexfrabdioph  36268  4rexfrabdioph  36270
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