ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  opabbid GIF version

Theorem opabbid 3988
Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (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 459 . . . . 5 (𝜑 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)))
52, 4exbid 1595 . . . 4 (𝜑 → (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)))
61, 5exbid 1595 . . 3 (𝜑 → (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)))
76abbidv 2255 . 2 (𝜑 → {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)})
8 df-opab 3985 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
9 df-opab 3985 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜒} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)}
107, 8, 93eqtr4g 2195 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wnf 1436  wex 1468  {cab 2123  cop 3525  {copab 3983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-opab 3985
This theorem is referenced by:  opabbidv  3989  mpteq12f  4003  fnoprabg  5865
  Copyright terms: Public domain W3C validator