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Theorem ifexd 4574
Description: Existence of the conditional operator (deduction form). (Contributed by SN, 26-Jul-2024.)
Hypotheses
Ref Expression
ifexd.1 (𝜑𝐴𝑉)
ifexd.2 (𝜑𝐵𝑊)
Assertion
Ref Expression
ifexd (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ V)

Proof of Theorem ifexd
StepHypRef Expression
1 ifexd.1 . . 3 (𝜑𝐴𝑉)
21elexd 3504 . 2 (𝜑𝐴 ∈ V)
3 ifexd.2 . . 3 (𝜑𝐵𝑊)
43elexd 3504 . 2 (𝜑𝐵 ∈ V)
52, 4ifcld 4572 1 (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  Vcvv 3480  ifcif 4525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-if 4526
This theorem is referenced by:  ifexg  4575  evlslem3  22104  mhpsclcl  22151  psgnfzto1stlem  33120  prjspnfv01  42634  prjspner01  42635  prjspner1  42636  sge0val  46381  hsphoival  46594  hspmbllem2  46642
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