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Theorem ssopab2dv 3978
Description: Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 19-Jan-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Hypothesis
Ref Expression
ssopab2dv.1 (φ → (ψχ))
Assertion
Ref Expression
ssopab2dv (φ → {⟨x, y⟩ ∣ ψ} ⊆ {⟨x, y⟩ ∣ χ})
Distinct variable groups:   φ,x   φ,y
Allowed substitution hints:   ψ(x,y)   χ(x,y)

Proof of Theorem ssopab2dv
StepHypRef Expression
1 ssopab2dv.1 . . 3 (φ → (ψχ))
21alrimivv 1728 . 2 (φxy(ψχ))
3 ssopab2 3975 . 2 (xy(ψχ) → {⟨x, y⟩ ∣ ψ} ⊆ {⟨x, y⟩ ∣ χ})
42, 3syl 14 1 (φ → {⟨x, y⟩ ∣ ψ} ⊆ {⟨x, y⟩ ∣ χ})
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1221  wss 2885  {copab 3780
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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-in 2892  df-ss 2899  df-opab 3782
This theorem is referenced by:  xpss12  4360  coss1  4406  coss2  4407  cnvss  4423
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