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Theorem ssopab2 4003
 Description: Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 19-May-2013.)
Assertion
Ref Expression
ssopab2 (xy(φψ) → {⟨x, y⟩ ∣ φ} ⊆ {⟨x, y⟩ ∣ ψ})

Proof of Theorem ssopab2
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 nfa1 1431 . . . 4 xxy(φψ)
2 nfa1 1431 . . . . . 6 yy(φψ)
3 sp 1398 . . . . . . 7 (y(φψ) → (φψ))
43anim2d 320 . . . . . 6 (y(φψ) → ((z = ⟨x, y φ) → (z = ⟨x, y ψ)))
52, 4eximd 1500 . . . . 5 (y(φψ) → (y(z = ⟨x, y φ) → y(z = ⟨x, y ψ)))
65sps 1427 . . . 4 (xy(φψ) → (y(z = ⟨x, y φ) → y(z = ⟨x, y ψ)))
71, 6eximd 1500 . . 3 (xy(φψ) → (xy(z = ⟨x, y φ) → xy(z = ⟨x, y ψ)))
87ss2abdv 3007 . 2 (xy(φψ) → {zxy(z = ⟨x, y φ)} ⊆ {zxy(z = ⟨x, y ψ)})
9 df-opab 3810 . 2 {⟨x, y⟩ ∣ φ} = {zxy(z = ⟨x, y φ)}
10 df-opab 3810 . 2 {⟨x, y⟩ ∣ ψ} = {zxy(z = ⟨x, y ψ)}
118, 9, 103sstr4g 2980 1 (xy(φψ) → {⟨x, y⟩ ∣ φ} ⊆ {⟨x, y⟩ ∣ ψ})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240   = wceq 1242  ∃wex 1378  {cab 2023   ⊆ wss 2911  ⟨cop 3370  {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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925  df-opab 3810 This theorem is referenced by:  ssopab2b  4004  ssopab2i  4005  ssopab2dv  4006
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