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Theorem axxpprim 4090
 Description: ax-xp 4079 presented without any set theory definitions. (Contributed by SF, 25-Mar-2015.)
Assertion
Ref Expression
axxpprim yz(z ywt(a(a z ↔ (b(b ab = w) c(c a ↔ (c = w c = t)))) t x))
Distinct variable groups:   a,b   a,c,t,w   z,a   w,b   t,c,w   x,t,y,z,w

Proof of Theorem axxpprim
StepHypRef Expression
1 ax-xp 4079 . 2 yz(z ywt(z = ⟪w, t t x))
2 axprimlem2 4089 . . . . . . 7 (z = ⟪w, t⟫ ↔ a(a z ↔ (b(b ab = w) c(c a ↔ (c = w c = t)))))
32anbi1i 676 . . . . . 6 ((z = ⟪w, t t x) ↔ (a(a z ↔ (b(b ab = w) c(c a ↔ (c = w c = t)))) t x))
432exbii 1583 . . . . 5 (wt(z = ⟪w, t t x) ↔ wt(a(a z ↔ (b(b ab = w) c(c a ↔ (c = w c = t)))) t x))
54bibi2i 304 . . . 4 ((z ywt(z = ⟪w, t t x)) ↔ (z ywt(a(a z ↔ (b(b ab = w) c(c a ↔ (c = w c = t)))) t x)))
65albii 1566 . . 3 (z(z ywt(z = ⟪w, t t x)) ↔ z(z ywt(a(a z ↔ (b(b ab = w) c(c a ↔ (c = w c = t)))) t x)))
76exbii 1582 . 2 (yz(z ywt(z = ⟪w, t t x)) ↔ yz(z ywt(a(a z ↔ (b(b ab = w) c(c a ↔ (c = w c = t)))) t x)))
81, 7mpbi 199 1 yz(z ywt(a(a z ↔ (b(b ab = w) c(c a ↔ (c = w c = t)))) t x))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642  ⟪copk 4057 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  ax-xp 4079 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-un 3214  df-sn 3741  df-pr 3742  df-opk 4058 This theorem is referenced by: (None)
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