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Theorem eqopab2b 3990
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 3987 . . 3 ({⟨x, y⟩ ∣ φ} ⊆ {⟨x, y⟩ ∣ ψ} ↔ xy(φψ))
2 ssopab2b 3987 . . 3 ({⟨x, y⟩ ∣ ψ} ⊆ {⟨x, y⟩ ∣ φ} ↔ xy(ψφ))
31, 2anbi12i 436 . 2 (({⟨x, y⟩ ∣ φ} ⊆ {⟨x, y⟩ ∣ ψ} {⟨x, y⟩ ∣ ψ} ⊆ {⟨x, y⟩ ∣ φ}) ↔ (xy(φψ) xy(ψφ)))
4 eqss 2937 . 2 ({⟨x, y⟩ ∣ φ} = {⟨x, y⟩ ∣ ψ} ↔ ({⟨x, y⟩ ∣ φ} ⊆ {⟨x, y⟩ ∣ ψ} {⟨x, y⟩ ∣ ψ} ⊆ {⟨x, y⟩ ∣ φ}))
5 2albiim 1358 . 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 1226   = wceq 1228  wss 2894  {copab 3791
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-opab 3793
This theorem is referenced by: (None)
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