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Theorem oprabbidv 7209
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
StepHypRef Expression
1 nfv 1906 . 2 𝑥𝜑
2 nfv 1906 . 2 𝑦𝜑
3 nfv 1906 . 2 𝑧𝜑
4 oprabbidv.1 . 2 (𝜑 → (𝜓𝜒))
51, 2, 3, 4oprabbid 7208 1 (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  {coprab 7146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-9 2115  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-oprab 7149
This theorem is referenced by:  oprabbii  7210  mpoeq123dva  7217  mpoeq3dva  7220  resoprab2  7260  erovlem  8382  joinfval  17599  meetfval  17613  odumeet  17738  odujoin  17740  mppsval  32716  csbmpo123  34494  unceq  34750  uncf  34752  unccur  34756
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