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

Theorem oprabbii 6608
 Description: Equivalent wff's yield equal operation class abstractions. (Contributed by NM, 28-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
Hypothesis
Ref Expression
oprabbii.1 (𝜑𝜓)
Assertion
Ref Expression
oprabbii {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem oprabbii
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . 2 𝑤 = 𝑤
2 oprabbii.1 . . . 4 (𝜑𝜓)
32a1i 11 . . 3 (𝑤 = 𝑤 → (𝜑𝜓))
43oprabbidv 6607 . 2 (𝑤 = 𝑤 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓})
51, 4ax-mp 5 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475  {coprab 6550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-oprab 6553 This theorem is referenced by:  oprab4  6624  mpt2v  6648  dfxp3  7119  tposmpt2  7276  addsrpr  9775  mulsrpr  9776  addcnsr  9835  mulcnsr  9836  joinfval2  16825  meetfval2  16839
 Copyright terms: Public domain W3C validator