Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  oprabbidv Structured version   Visualization version   GIF version

Theorem oprabbidv 7220
 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 1915 . 2 𝑥𝜑
2 nfv 1915 . 2 𝑦𝜑
3 nfv 1915 . 2 𝑧𝜑
4 oprabbidv.1 . 2 (𝜑 → (𝜓𝜒))
51, 2, 3, 4oprabbid 7219 1 (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜒})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538  {coprab 7157 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-9 2121  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-oprab 7160 This theorem is referenced by:  oprabbii  7221  mpoeq123dva  7228  mpoeq3dva  7231  resoprab2  7271  erovlem  8409  joinfval  17690  meetfval  17704  odumeet  17829  odujoin  17831  mppsval  33062  csbmpo123  35062  unceq  35348  uncf  35350  unccur  35354
 Copyright terms: Public domain W3C validator