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Theorem ifeq12da 4523
Description: Equivalence deduction for conditional operators. (Contributed by Wolf Lammen, 24-Jun-2021.)
Hypotheses
Ref Expression
ifeq12da.1 ((𝜑𝜓) → 𝐴 = 𝐶)
ifeq12da.2 ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)
Assertion
Ref Expression
ifeq12da (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))

Proof of Theorem ifeq12da
StepHypRef Expression
1 ifeq12da.1 . . . 4 ((𝜑𝜓) → 𝐴 = 𝐶)
21ifeq1da 4521 . . 3 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐵))
3 iftrue 4496 . . . 4 (𝜓 → if(𝜓, 𝐶, 𝐵) = 𝐶)
4 iftrue 4496 . . . 4 (𝜓 → if(𝜓, 𝐶, 𝐷) = 𝐶)
53, 4eqtr4d 2776 . . 3 (𝜓 → if(𝜓, 𝐶, 𝐵) = if(𝜓, 𝐶, 𝐷))
62, 5sylan9eq 2793 . 2 ((𝜑𝜓) → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
7 ifeq12da.2 . . . 4 ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)
87ifeq2da 4522 . . 3 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐷))
9 iffalse 4499 . . . 4 𝜓 → if(𝜓, 𝐴, 𝐷) = 𝐷)
10 iffalse 4499 . . . 4 𝜓 → if(𝜓, 𝐶, 𝐷) = 𝐷)
119, 10eqtr4d 2776 . . 3 𝜓 → if(𝜓, 𝐴, 𝐷) = if(𝜓, 𝐶, 𝐷))
128, 11sylan9eq 2793 . 2 ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
136, 12pm2.61dan 812 1 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  ifcif 4490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3449  df-un 3919  df-if 4491
This theorem is referenced by:  ifbieq12d2  4524  copco  24404  pcohtpylem  24405  rpvmasum2  26883  prjspnfv01  41009
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