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Theorem elimhyps2 33067
Description: Generalization of elimhyps 33064 that is not useful unless we can separately prove 𝐴 ∈ V. (Contributed by NM, 13-Jun-2019.)
Hypothesis
Ref Expression
elimhyps2.1 [𝐵 / 𝑥]𝜑
Assertion
Ref Expression
elimhyps2 [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑

Proof of Theorem elimhyps2
StepHypRef Expression
1 dfsbcq 3399 . 2 (𝐴 = if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) → ([𝐴 / 𝑥]𝜑[if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑))
2 dfsbcq 3399 . 2 (𝐵 = if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) → ([𝐵 / 𝑥]𝜑[if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑))
3 elimhyps2.1 . 2 [𝐵 / 𝑥]𝜑
41, 2, 3elimhyp 4091 1 [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑
Colors of variables: wff setvar class
Syntax hints:  [wsbc 3397  ifcif 4031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-sbc 3398  df-if 4032
This theorem is referenced by: (None)
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