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Theorem supeq123d 7004
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 4026 . . . . . . 7 (𝜑 → (𝑥𝐶𝑦𝑥𝐹𝑦))
54notbid 668 . . . . . 6 (𝜑 → (¬ 𝑥𝐶𝑦 ↔ ¬ 𝑥𝐹𝑦))
62, 5raleqbidv 2695 . . . . 5 (𝜑 → (∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ↔ ∀𝑦𝐷 ¬ 𝑥𝐹𝑦))
73breqd 4026 . . . . . . 7 (𝜑 → (𝑦𝐶𝑥𝑦𝐹𝑥))
83breqd 4026 . . . . . . . 8 (𝜑 → (𝑦𝐶𝑧𝑦𝐹𝑧))
92, 8rexeqbidv 2696 . . . . . . 7 (𝜑 → (∃𝑧𝐴 𝑦𝐶𝑧 ↔ ∃𝑧𝐷 𝑦𝐹𝑧))
107, 9imbi12d 234 . . . . . 6 (𝜑 → ((𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧) ↔ (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧)))
111, 10raleqbidv 2695 . . . . 5 (𝜑 → (∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧) ↔ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧)))
126, 11anbi12d 473 . . . 4 (𝜑 → ((∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧)) ↔ (∀𝑦𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧))))
131, 12rabeqbidv 2744 . . 3 (𝜑 → {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧))} = {𝑥𝐸 ∣ (∀𝑦𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧))})
1413unieqd 3832 . 2 (𝜑 {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧))} = {𝑥𝐸 ∣ (∀𝑦𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧))})
15 df-sup 6997 . 2 sup(𝐴, 𝐵, 𝐶) = {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦𝐵 (𝑦𝐶𝑥 → ∃𝑧𝐴 𝑦𝐶𝑧))}
16 df-sup 6997 . 2 sup(𝐷, 𝐸, 𝐹) = {𝑥𝐸 ∣ (∀𝑦𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦𝐸 (𝑦𝐹𝑥 → ∃𝑧𝐷 𝑦𝐹𝑧))}
1714, 15, 163eqtr4g 2245 1 (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104   = wceq 1363  wral 2465  wrex 2466  {crab 2469   cuni 3821   class class class wbr 4015  supcsup 6995
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-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-uni 3822  df-br 4016  df-sup 6997
This theorem is referenced by:  infeq123d  7029
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