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Theorem eqifdc 3584
Description: Expansion of an equality with a conditional operator. (Contributed by Jim Kingdon, 28-Jul-2022.)
Assertion
Ref Expression
eqifdc (DECID 𝜑 → (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴 = 𝐵) ∨ (¬ 𝜑𝐴 = 𝐶))))

Proof of Theorem eqifdc
StepHypRef Expression
1 exmiddc 837 . . 3 (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
2 simpr 110 . . . . . 6 ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → 𝜑)
3 simpl 109 . . . . . . 7 ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → 𝐴 = if(𝜑, 𝐵, 𝐶))
42iftrued 3556 . . . . . . 7 ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → if(𝜑, 𝐵, 𝐶) = 𝐵)
53, 4eqtrd 2222 . . . . . 6 ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → 𝐴 = 𝐵)
62, 5jca 306 . . . . 5 ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → (𝜑𝐴 = 𝐵))
76ex 115 . . . 4 (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝜑 → (𝜑𝐴 = 𝐵)))
8 simpr 110 . . . . . 6 ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → ¬ 𝜑)
9 simpl 109 . . . . . . 7 ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → 𝐴 = if(𝜑, 𝐵, 𝐶))
108iffalsed 3559 . . . . . . 7 ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → if(𝜑, 𝐵, 𝐶) = 𝐶)
119, 10eqtrd 2222 . . . . . 6 ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → 𝐴 = 𝐶)
128, 11jca 306 . . . . 5 ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → (¬ 𝜑𝐴 = 𝐶))
1312ex 115 . . . 4 (𝐴 = if(𝜑, 𝐵, 𝐶) → (¬ 𝜑 → (¬ 𝜑𝐴 = 𝐶)))
147, 13orim12d 787 . . 3 (𝐴 = if(𝜑, 𝐵, 𝐶) → ((𝜑 ∨ ¬ 𝜑) → ((𝜑𝐴 = 𝐵) ∨ (¬ 𝜑𝐴 = 𝐶))))
151, 14syl5com 29 . 2 (DECID 𝜑 → (𝐴 = if(𝜑, 𝐵, 𝐶) → ((𝜑𝐴 = 𝐵) ∨ (¬ 𝜑𝐴 = 𝐶))))
16 simpr 110 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐴 = 𝐵)
17 simpl 109 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝜑)
1817iftrued 3556 . . . 4 ((𝜑𝐴 = 𝐵) → if(𝜑, 𝐵, 𝐶) = 𝐵)
1916, 18eqtr4d 2225 . . 3 ((𝜑𝐴 = 𝐵) → 𝐴 = if(𝜑, 𝐵, 𝐶))
20 simpr 110 . . . 4 ((¬ 𝜑𝐴 = 𝐶) → 𝐴 = 𝐶)
21 simpl 109 . . . . 5 ((¬ 𝜑𝐴 = 𝐶) → ¬ 𝜑)
2221iffalsed 3559 . . . 4 ((¬ 𝜑𝐴 = 𝐶) → if(𝜑, 𝐵, 𝐶) = 𝐶)
2320, 22eqtr4d 2225 . . 3 ((¬ 𝜑𝐴 = 𝐶) → 𝐴 = if(𝜑, 𝐵, 𝐶))
2419, 23jaoi 717 . 2 (((𝜑𝐴 = 𝐵) ∨ (¬ 𝜑𝐴 = 𝐶)) → 𝐴 = if(𝜑, 𝐵, 𝐶))
2515, 24impbid1 142 1 (DECID 𝜑 → (𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴 = 𝐵) ∨ (¬ 𝜑𝐴 = 𝐶))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 709  DECID wdc 835   = wceq 1364  ifcif 3549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-dc 836  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-if 3550
This theorem is referenced by:  fodjum  7175  nninfwlporlemd  7201  xrmaxiflemcom  11292  subctctexmid  15229
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