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

Theorem opabbid 4847
 Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Hypotheses
Ref Expression
opabbid.1 𝑥𝜑
opabbid.2 𝑦𝜑
opabbid.3 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
opabbid (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})

Proof of Theorem opabbid
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 opabbid.1 . . . 4 𝑥𝜑
2 opabbid.2 . . . . 5 𝑦𝜑
3 opabbid.3 . . . . . 6 (𝜑 → (𝜓𝜒))
43anbi2d 606 . . . . 5 (𝜑 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)))
52, 4exbid 2246 . . . 4 (𝜑 → (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)))
61, 5exbid 2246 . . 3 (𝜑 → (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)))
76abbidv 2889 . 2 (𝜑 → {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)})
8 df-opab 4845 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
9 df-opab 4845 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜒} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)}
107, 8, 93eqtr4g 2829 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1630  ∃wex 1851  Ⅎwnf 1855  {cab 2756  ⟨cop 4320  {copab 4844 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-opab 4845 This theorem is referenced by:  opabbidv  4848  mpteq12f  4863  mpteq12d  4866  mpteq12df  4867  feqmptdf  6393  fnoprabg  6907  sprsymrelfo  42265
 Copyright terms: Public domain W3C validator