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Theorem supeq123d 8913
 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 5044 . . . . . . 7 (𝜑 → (𝑥𝐶𝑦𝑥𝐹𝑦))
54notbid 321 . . . . . 6 (𝜑 → (¬ 𝑥𝐶𝑦 ↔ ¬ 𝑥𝐹𝑦))
62, 5raleqbidv 3354 . . . . 5 (𝜑 → (∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ↔ ∀𝑦𝐷 ¬ 𝑥𝐹𝑦))
73breqd 5044 . . . . . . 7 (𝜑 → (𝑦𝐶𝑥𝑦𝐹𝑥))
83breqd 5044 . . . . . . . 8 (𝜑 → (𝑦𝐶𝑧𝑦𝐹𝑧))
92, 8rexeqbidv 3355 . . . . . . 7 (𝜑 → (∃𝑧𝐴 𝑦𝐶𝑧 ↔ ∃𝑧𝐷 𝑦𝐹𝑧))
107, 9imbi12d 348 . . . . . 6 (𝜑 → ((𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧) ↔ (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧)))
111, 10raleqbidv 3354 . . . . 5 (𝜑 → (∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧) ↔ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧)))
126, 11anbi12d 633 . . . 4 (𝜑 → ((∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧)) ↔ (∀𝑦𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧))))
131, 12rabeqbidv 3433 . . 3 (𝜑 → {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧))} = {𝑥𝐸 ∣ (∀𝑦𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧))})
1413unieqd 4817 . 2 (𝜑 {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧))} = {𝑥𝐸 ∣ (∀𝑦𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧))})
15 df-sup 8905 . 2 sup(𝐴, 𝐵, 𝐶) = {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧))}
16 df-sup 8905 . 2 sup(𝐷, 𝐸, 𝐹) = {𝑥𝐸 ∣ (∀𝑦𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧))}
1714, 15, 163eqtr4g 2858 1 (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538  ∀wral 3106  ∃wrex 3107  {crab 3110  ∪ cuni 4803   class class class wbr 5033  supcsup 8903 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-in 3889  df-ss 3899  df-uni 4804  df-br 5034  df-sup 8905 This theorem is referenced by:  infeq123d  8944  wlimeq12  33282  aomclem8  40092
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