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Theorem ifeq3da 29670
 Description: Given an expression 𝐶 containing if(𝜓, 𝐸, 𝐹), substitute (hypotheses .1 and .2) and evaluate (hypotheses .3 and .4) it for both cases at the same time. (Contributed by Thierry Arnoux, 13-Dec-2021.)
Hypotheses
Ref Expression
ifeq3da.1 (if(𝜓, 𝐸, 𝐹) = 𝐸𝐶 = 𝐺)
ifeq3da.2 (if(𝜓, 𝐸, 𝐹) = 𝐹𝐶 = 𝐻)
ifeq3da.3 (𝜑𝐺 = 𝐴)
ifeq3da.4 (𝜑𝐻 = 𝐵)
Assertion
Ref Expression
ifeq3da (𝜑 → if(𝜓, 𝐴, 𝐵) = 𝐶)

Proof of Theorem ifeq3da
StepHypRef Expression
1 iftrue 4234 . . . . 5 (𝜓 → if(𝜓, 𝐸, 𝐹) = 𝐸)
2 ifeq3da.1 . . . . 5 (if(𝜓, 𝐸, 𝐹) = 𝐸𝐶 = 𝐺)
31, 2syl 17 . . . 4 (𝜓𝐶 = 𝐺)
43adantl 473 . . 3 ((𝜑𝜓) → 𝐶 = 𝐺)
5 ifeq3da.3 . . . 4 (𝜑𝐺 = 𝐴)
65adantr 472 . . 3 ((𝜑𝜓) → 𝐺 = 𝐴)
74, 6eqtr2d 2793 . 2 ((𝜑𝜓) → 𝐴 = 𝐶)
8 iffalse 4237 . . . . 5 𝜓 → if(𝜓, 𝐸, 𝐹) = 𝐹)
9 ifeq3da.2 . . . . 5 (if(𝜓, 𝐸, 𝐹) = 𝐹𝐶 = 𝐻)
108, 9syl 17 . . . 4 𝜓𝐶 = 𝐻)
1110adantl 473 . . 3 ((𝜑 ∧ ¬ 𝜓) → 𝐶 = 𝐻)
12 ifeq3da.4 . . . 4 (𝜑𝐻 = 𝐵)
1312adantr 472 . . 3 ((𝜑 ∧ ¬ 𝜓) → 𝐻 = 𝐵)
1411, 13eqtr2d 2793 . 2 ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐶)
157, 14ifeqda 4263 1 (𝜑 → if(𝜓, 𝐴, 𝐵) = 𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1630  ifcif 4228 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-if 4229 This theorem is referenced by:  circlemeth  31025
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