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Theorem eqifdc 3506
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 821 . . 3 (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
2 simpr 109 . . . . . 6 ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → 𝜑)
3 simpl 108 . . . . . . 7 ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → 𝐴 = if(𝜑, 𝐵, 𝐶))
42iftrued 3481 . . . . . . 7 ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → if(𝜑, 𝐵, 𝐶) = 𝐵)
53, 4eqtrd 2172 . . . . . 6 ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → 𝐴 = 𝐵)
62, 5jca 304 . . . . 5 ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ 𝜑) → (𝜑𝐴 = 𝐵))
76ex 114 . . . 4 (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝜑 → (𝜑𝐴 = 𝐵)))
8 simpr 109 . . . . . 6 ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → ¬ 𝜑)
9 simpl 108 . . . . . . 7 ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → 𝐴 = if(𝜑, 𝐵, 𝐶))
108iffalsed 3484 . . . . . . 7 ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → if(𝜑, 𝐵, 𝐶) = 𝐶)
119, 10eqtrd 2172 . . . . . 6 ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → 𝐴 = 𝐶)
128, 11jca 304 . . . . 5 ((𝐴 = if(𝜑, 𝐵, 𝐶) ∧ ¬ 𝜑) → (¬ 𝜑𝐴 = 𝐶))
1312ex 114 . . . 4 (𝐴 = if(𝜑, 𝐵, 𝐶) → (¬ 𝜑 → (¬ 𝜑𝐴 = 𝐶)))
147, 13orim12d 775 . . 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 109 . . . 4 ((𝜑𝐴 = 𝐵) → 𝐴 = 𝐵)
17 simpl 108 . . . . 5 ((𝜑𝐴 = 𝐵) → 𝜑)
1817iftrued 3481 . . . 4 ((𝜑𝐴 = 𝐵) → if(𝜑, 𝐵, 𝐶) = 𝐵)
1916, 18eqtr4d 2175 . . 3 ((𝜑𝐴 = 𝐵) → 𝐴 = if(𝜑, 𝐵, 𝐶))
20 simpr 109 . . . 4 ((¬ 𝜑𝐴 = 𝐶) → 𝐴 = 𝐶)
21 simpl 108 . . . . 5 ((¬ 𝜑𝐴 = 𝐶) → ¬ 𝜑)
2221iffalsed 3484 . . . 4 ((¬ 𝜑𝐴 = 𝐶) → if(𝜑, 𝐵, 𝐶) = 𝐶)
2320, 22eqtr4d 2175 . . 3 ((¬ 𝜑𝐴 = 𝐶) → 𝐴 = if(𝜑, 𝐵, 𝐶))
2419, 23jaoi 705 . 2 (((𝜑𝐴 = 𝐵) ∨ (¬ 𝜑𝐴 = 𝐶)) → 𝐴 = if(𝜑, 𝐵, 𝐶))
2515, 24impbid1 141 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 103  wb 104  wo 697  DECID wdc 819   = wceq 1331  ifcif 3474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-dc 820  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-if 3475
This theorem is referenced by:  fodjum  7018  xrmaxiflemcom  11030  subctctexmid  13255
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