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Theorem ssopab2 4712
 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 1788 . . . 4 xxy(φψ)
2 nfa1 1788 . . . . . 6 yy(φψ)
3 sp 1747 . . . . . . 7 (y(φψ) → (φψ))
43anim2d 548 . . . . . 6 (y(φψ) → ((z = x, y φ) → (z = x, y ψ)))
52, 4eximd 1770 . . . . 5 (y(φψ) → (y(z = x, y φ) → y(z = x, y ψ)))
65sps 1754 . . . 4 (xy(φψ) → (y(z = x, y φ) → y(z = x, y ψ)))
71, 6eximd 1770 . . 3 (xy(φψ) → (xy(z = x, y φ) → xy(z = x, y ψ)))
87ss2abdv 3339 . 2 (xy(φψ) → {z xy(z = x, y φ)} {z xy(z = x, y ψ)})
9 df-opab 4623 . 2 {x, y φ} = {z xy(z = x, y φ)}
10 df-opab 4623 . 2 {x, y ψ} = {z xy(z = x, y ψ)}
118, 9, 103sstr4g 3312 1 (xy(φψ) → {x, y φ} {x, y ψ})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642  {cab 2339   ⊆ wss 3257  ⟨cop 4561  {copab 4622 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-opab 4623 This theorem is referenced by:  ssopab2b  4713  ssopab2i  4714  ssopab2dv  4715
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