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Theorem oprabbidv 7477
Description: Equivalent wff's yield equal operation class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.)
Hypothesis
Ref Expression
oprabbidv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
oprabbidv (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒})
Distinct variable groups:   𝑥,𝑧,𝜑   𝑦,𝑧,𝜑
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑥,𝑦,𝑧)

Proof of Theorem oprabbidv
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 oprabbidv.1 . . . . . . 7 (𝜑 → (𝜓𝜒))
21anbi2d 627 . . . . . 6 (𝜑 → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒)))
32exbidv 1922 . . . . 5 (𝜑 → (∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒)))
43exbidv 1922 . . . 4 (𝜑 → (∃𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒)))
54exbidv 1922 . . 3 (𝜑 → (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒)))
65abbidv 2799 . 2 (𝜑 → {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒)})
7 df-oprab 7415 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)}
8 df-oprab 7415 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜒)}
96, 7, 83eqtr4g 2795 1 (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wex 1779  {cab 2707  cop 4633  {coprab 7412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-oprab 7415
This theorem is referenced by:  oprabbii  7478  mpoeq123dva  7485  mpoeq3dva  7488  resoprab2  7529  erovlem  8809  joinfval  18330  meetfval  18344  odujoin  18365  odumeet  18367  mppsval  34861  csbmpo123  36515  unceq  36768  uncf  36770  unccur  36774
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