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Theorem supeq123d 8709
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 4940 . . . . . . 7 (𝜑 → (𝑥𝐶𝑦𝑥𝐹𝑦))
54notbid 310 . . . . . 6 (𝜑 → (¬ 𝑥𝐶𝑦 ↔ ¬ 𝑥𝐹𝑦))
62, 5raleqbidv 3341 . . . . 5 (𝜑 → (∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ↔ ∀𝑦𝐷 ¬ 𝑥𝐹𝑦))
73breqd 4940 . . . . . . 7 (𝜑 → (𝑦𝐶𝑥𝑦𝐹𝑥))
83breqd 4940 . . . . . . . 8 (𝜑 → (𝑦𝐶𝑧𝑦𝐹𝑧))
92, 8rexeqbidv 3342 . . . . . . 7 (𝜑 → (∃𝑧𝐴 𝑦𝐶𝑧 ↔ ∃𝑧𝐷 𝑦𝐹𝑧))
107, 9imbi12d 337 . . . . . 6 (𝜑 → ((𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧) ↔ (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧)))
111, 10raleqbidv 3341 . . . . 5 (𝜑 → (∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧) ↔ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧)))
126, 11anbi12d 621 . . . 4 (𝜑 → ((∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧)) ↔ (∀𝑦𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧))))
131, 12rabeqbidv 3408 . . 3 (𝜑 → {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧))} = {𝑥𝐸 ∣ (∀𝑦𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧))})
1413unieqd 4722 . 2 (𝜑 {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧))} = {𝑥𝐸 ∣ (∀𝑦𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧))})
15 df-sup 8701 . 2 sup(𝐴, 𝐵, 𝐶) = {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧))}
16 df-sup 8701 . 2 sup(𝐷, 𝐸, 𝐹) = {𝑥𝐸 ∣ (∀𝑦𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧))}
1714, 15, 163eqtr4g 2839 1 (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387   = wceq 1507  wral 3088  wrex 3089  {crab 3092   cuni 4712   class class class wbr 4929  supcsup 8699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-ral 3093  df-rex 3094  df-rab 3097  df-uni 4713  df-br 4930  df-sup 8701
This theorem is referenced by:  infeq123d  8740  wlimeq12  32633  aomclem8  39063
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