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Theorem ssopab2 4040
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 1450 . . . 4 𝑥𝑥𝑦(𝜑𝜓)
2 nfa1 1450 . . . . . 6 𝑦𝑦(𝜑𝜓)
3 sp 1417 . . . . . . 7 (∀𝑦(𝜑𝜓) → (𝜑𝜓))
43anim2d 324 . . . . . 6 (∀𝑦(𝜑𝜓) → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
52, 4eximd 1519 . . . . 5 (∀𝑦(𝜑𝜓) → (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
65sps 1446 . . . 4 (∀𝑥𝑦(𝜑𝜓) → (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
71, 6eximd 1519 . . 3 (∀𝑥𝑦(𝜑𝜓) → (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
87ss2abdv 3041 . 2 (∀𝑥𝑦(𝜑𝜓) → {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ⊆ {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)})
9 df-opab 3847 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
10 df-opab 3847 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
118, 9, 103sstr4g 3014 1 (∀𝑥𝑦(𝜑𝜓) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1257   = wceq 1259  wex 1397  {cab 2042  wss 2945  cop 3406  {copab 3845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-in 2952  df-ss 2959  df-opab 3847
This theorem is referenced by:  ssopab2b  4041  ssopab2i  4042  ssopab2dv  4043
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