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Theorem 2sbcrexOLD 38136
Description: Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 6919 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
2sbcrex.1 𝐴 ∈ V
2sbcrex.2 𝐵 ∈ V
Assertion
Ref Expression
2sbcrexOLD ([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 2sbcrexOLD
StepHypRef Expression
1 2sbcrex.2 . . . 4 𝐵 ∈ V
2 sbcrexgOLD 38135 . . . 4 (𝐵 ∈ V → ([𝐵 / 𝑏]𝑐𝐶 𝜑 ↔ ∃𝑐𝐶 [𝐵 / 𝑏]𝜑))
31, 2ax-mp 5 . . 3 ([𝐵 / 𝑏]𝑐𝐶 𝜑 ↔ ∃𝑐𝐶 [𝐵 / 𝑏]𝜑)
43sbcbii 3689 . 2 ([𝐴 / 𝑎][𝐵 / 𝑏]𝑐𝐶 𝜑[𝐴 / 𝑎]𝑐𝐶 [𝐵 / 𝑏]𝜑)
5 2sbcrex.1 . . 3 𝐴 ∈ V
6 sbcrexgOLD 38135 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑎]𝑐𝐶 [𝐵 / 𝑏]𝜑 ↔ ∃𝑐𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑))
75, 6ax-mp 5 . 2 ([𝐴 / 𝑎]𝑐𝐶 [𝐵 / 𝑏]𝜑 ↔ ∃𝑐𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑)
84, 7bitri 267 1 ([𝐴 / 𝑎][𝐵 / 𝑏]𝑐𝐶 𝜑 ↔ ∃𝑐𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wcel 2157  wrex 3090  Vcvv 3385  [wsbc 3633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-v 3387  df-sbc 3634
This theorem is referenced by: (None)
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