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Theorem supeq123d 8517
Description: Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Hypotheses
Ref Expression
supeq123d.a (𝜑𝐴 = 𝐷)
supeq123d.b (𝜑𝐵 = 𝐸)
supeq123d.c (𝜑𝐶 = 𝐹)
Assertion
Ref Expression
supeq123d (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹))

Proof of Theorem supeq123d
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supeq123d.b . . . 4 (𝜑𝐵 = 𝐸)
2 supeq123d.a . . . . . 6 (𝜑𝐴 = 𝐷)
3 supeq123d.c . . . . . . . 8 (𝜑𝐶 = 𝐹)
43breqd 4811 . . . . . . 7 (𝜑 → (𝑥𝐶𝑦𝑥𝐹𝑦))
54notbid 307 . . . . . 6 (𝜑 → (¬ 𝑥𝐶𝑦 ↔ ¬ 𝑥𝐹𝑦))
62, 5raleqbidv 3287 . . . . 5 (𝜑 → (∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ↔ ∀𝑦𝐷 ¬ 𝑥𝐹𝑦))
73breqd 4811 . . . . . . 7 (𝜑 → (𝑦𝐶𝑥𝑦𝐹𝑥))
83breqd 4811 . . . . . . . 8 (𝜑 → (𝑦𝐶𝑧𝑦𝐹𝑧))
92, 8rexeqbidv 3288 . . . . . . 7 (𝜑 → (∃𝑧𝐴 𝑦𝐶𝑧 ↔ ∃𝑧𝐷 𝑦𝐹𝑧))
107, 9imbi12d 333 . . . . . 6 (𝜑 → ((𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧) ↔ (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧)))
111, 10raleqbidv 3287 . . . . 5 (𝜑 → (∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧) ↔ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧)))
126, 11anbi12d 749 . . . 4 (𝜑 → ((∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧)) ↔ (∀𝑦𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧))))
131, 12rabeqbidv 3331 . . 3 (𝜑 → {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧))} = {𝑥𝐸 ∣ (∀𝑦𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧))})
1413unieqd 4594 . 2 (𝜑 {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧))} = {𝑥𝐸 ∣ (∀𝑦𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧))})
15 df-sup 8509 . 2 sup(𝐴, 𝐵, 𝐶) = {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧))}
16 df-sup 8509 . 2 sup(𝐷, 𝐸, 𝐹) = {𝑥𝐸 ∣ (∀𝑦𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧))}
1714, 15, 163eqtr4g 2815 1 (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1628  wral 3046  wrex 3047  {crab 3050   cuni 4584   class class class wbr 4800  supcsup 8507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rex 3052  df-rab 3055  df-uni 4585  df-br 4801  df-sup 8509
This theorem is referenced by:  infeq123d  8548  wlimeq12  32066  aomclem8  38129
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