ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eqopab2b Structured version   GIF version

Theorem eqopab2b 4007
Description: Equivalence of ordered pair abstraction equality and biconditional. (Contributed by Mario Carneiro, 4-Jan-2017.)
Assertion
Ref Expression
eqopab2b ({⟨x, y⟩ ∣ φ} = {⟨x, y⟩ ∣ ψ} ↔ xy(φψ))

Proof of Theorem eqopab2b
StepHypRef Expression
1 ssopab2b 4004 . . 3 ({⟨x, y⟩ ∣ φ} ⊆ {⟨x, y⟩ ∣ ψ} ↔ xy(φψ))
2 ssopab2b 4004 . . 3 ({⟨x, y⟩ ∣ ψ} ⊆ {⟨x, y⟩ ∣ φ} ↔ xy(ψφ))
31, 2anbi12i 433 . 2 (({⟨x, y⟩ ∣ φ} ⊆ {⟨x, y⟩ ∣ ψ} {⟨x, y⟩ ∣ ψ} ⊆ {⟨x, y⟩ ∣ φ}) ↔ (xy(φψ) xy(ψφ)))
4 eqss 2954 . 2 ({⟨x, y⟩ ∣ φ} = {⟨x, y⟩ ∣ ψ} ↔ ({⟨x, y⟩ ∣ φ} ⊆ {⟨x, y⟩ ∣ ψ} {⟨x, y⟩ ∣ ψ} ⊆ {⟨x, y⟩ ∣ φ}))
5 2albiim 1374 . 2 (xy(φψ) ↔ (xy(φψ) xy(ψφ)))
63, 4, 53bitr4i 201 1 ({⟨x, y⟩ ∣ φ} = {⟨x, y⟩ ∣ ψ} ↔ xy(φψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242  wss 2911  {copab 3808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator