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Theorem ssoprab2 5503
Description: Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4003. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
Assertion
Ref Expression
ssoprab2 (xyz(φψ) → {⟨⟨x, y⟩, z⟩ ∣ φ} ⊆ {⟨⟨x, y⟩, z⟩ ∣ ψ})

Proof of Theorem ssoprab2
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 id 19 . . . . . . . . . 10 ((φψ) → (φψ))
21anim2d 320 . . . . . . . . 9 ((φψ) → ((w = ⟨⟨x, y⟩, z φ) → (w = ⟨⟨x, y⟩, z ψ)))
32alimi 1341 . . . . . . . 8 (z(φψ) → z((w = ⟨⟨x, y⟩, z φ) → (w = ⟨⟨x, y⟩, z ψ)))
4 exim 1487 . . . . . . . 8 (z((w = ⟨⟨x, y⟩, z φ) → (w = ⟨⟨x, y⟩, z ψ)) → (z(w = ⟨⟨x, y⟩, z φ) → z(w = ⟨⟨x, y⟩, z ψ)))
53, 4syl 14 . . . . . . 7 (z(φψ) → (z(w = ⟨⟨x, y⟩, z φ) → z(w = ⟨⟨x, y⟩, z ψ)))
65alimi 1341 . . . . . 6 (yz(φψ) → y(z(w = ⟨⟨x, y⟩, z φ) → z(w = ⟨⟨x, y⟩, z ψ)))
7 exim 1487 . . . . . 6 (y(z(w = ⟨⟨x, y⟩, z φ) → z(w = ⟨⟨x, y⟩, z ψ)) → (yz(w = ⟨⟨x, y⟩, z φ) → yz(w = ⟨⟨x, y⟩, z ψ)))
86, 7syl 14 . . . . 5 (yz(φψ) → (yz(w = ⟨⟨x, y⟩, z φ) → yz(w = ⟨⟨x, y⟩, z ψ)))
98alimi 1341 . . . 4 (xyz(φψ) → x(yz(w = ⟨⟨x, y⟩, z φ) → yz(w = ⟨⟨x, y⟩, z ψ)))
10 exim 1487 . . . 4 (x(yz(w = ⟨⟨x, y⟩, z φ) → yz(w = ⟨⟨x, y⟩, z ψ)) → (xyz(w = ⟨⟨x, y⟩, z φ) → xyz(w = ⟨⟨x, y⟩, z ψ)))
119, 10syl 14 . . 3 (xyz(φψ) → (xyz(w = ⟨⟨x, y⟩, z φ) → xyz(w = ⟨⟨x, y⟩, z ψ)))
1211ss2abdv 3007 . 2 (xyz(φψ) → {wxyz(w = ⟨⟨x, y⟩, z φ)} ⊆ {wxyz(w = ⟨⟨x, y⟩, z ψ)})
13 df-oprab 5459 . 2 {⟨⟨x, y⟩, z⟩ ∣ φ} = {wxyz(w = ⟨⟨x, y⟩, z φ)}
14 df-oprab 5459 . 2 {⟨⟨x, y⟩, z⟩ ∣ ψ} = {wxyz(w = ⟨⟨x, y⟩, z ψ)}
1512, 13, 143sstr4g 2980 1 (xyz(φψ) → {⟨⟨x, y⟩, z⟩ ∣ φ} ⊆ {⟨⟨x, y⟩, z⟩ ∣ 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  {coprab 5456
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-oprab 5459
This theorem is referenced by:  ssoprab2b  5504
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