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Theorem axtyplowerprim 4094
 Description: ax-typlower 4086 presented without any set theory definitions. (Contributed by SF, 25-Mar-2015.)
Assertion
Ref Expression
axtyplowerprim yz(z ywa(b(b a ↔ (c(c bc = w) d(d b ↔ (d = w e(e de = z))))) a x))
Distinct variable groups:   a,b,w   x,a,z   b,c   b,d,w,z   w,c   e,d   w,d,z   z,e   x,w,y,z

Proof of Theorem axtyplowerprim
StepHypRef Expression
1 ax-typlower 4086 . 2 yz(z yww, {z}⟫ x)
2 df-clel 2349 . . . . . . 7 (⟪w, {z}⟫ xa(a = ⟪w, {z}⟫ a x))
3 axprimlem2 4089 . . . . . . . . . 10 (a = ⟪w, {z}⟫ ↔ b(b a ↔ (c(c bc = w) d(d b ↔ (d = w d = {z})))))
4 axprimlem1 4088 . . . . . . . . . . . . . . . 16 (d = {z} ↔ e(e de = z))
54orbi2i 505 . . . . . . . . . . . . . . 15 ((d = w d = {z}) ↔ (d = w e(e de = z)))
65bibi2i 304 . . . . . . . . . . . . . 14 ((d b ↔ (d = w d = {z})) ↔ (d b ↔ (d = w e(e de = z))))
76albii 1566 . . . . . . . . . . . . 13 (d(d b ↔ (d = w d = {z})) ↔ d(d b ↔ (d = w e(e de = z))))
87orbi2i 505 . . . . . . . . . . . 12 ((c(c bc = w) d(d b ↔ (d = w d = {z}))) ↔ (c(c bc = w) d(d b ↔ (d = w e(e de = z)))))
98bibi2i 304 . . . . . . . . . . 11 ((b a ↔ (c(c bc = w) d(d b ↔ (d = w d = {z})))) ↔ (b a ↔ (c(c bc = w) d(d b ↔ (d = w e(e de = z))))))
109albii 1566 . . . . . . . . . 10 (b(b a ↔ (c(c bc = w) d(d b ↔ (d = w d = {z})))) ↔ b(b a ↔ (c(c bc = w) d(d b ↔ (d = w e(e de = z))))))
113, 10bitri 240 . . . . . . . . 9 (a = ⟪w, {z}⟫ ↔ b(b a ↔ (c(c bc = w) d(d b ↔ (d = w e(e de = z))))))
1211anbi1i 676 . . . . . . . 8 ((a = ⟪w, {z}⟫ a x) ↔ (b(b a ↔ (c(c bc = w) d(d b ↔ (d = w e(e de = z))))) a x))
1312exbii 1582 . . . . . . 7 (a(a = ⟪w, {z}⟫ a x) ↔ a(b(b a ↔ (c(c bc = w) d(d b ↔ (d = w e(e de = z))))) a x))
142, 13bitri 240 . . . . . 6 (⟪w, {z}⟫ xa(b(b a ↔ (c(c bc = w) d(d b ↔ (d = w e(e de = z))))) a x))
1514albii 1566 . . . . 5 (ww, {z}⟫ xwa(b(b a ↔ (c(c bc = w) d(d b ↔ (d = w e(e de = z))))) a x))
1615bibi2i 304 . . . 4 ((z yww, {z}⟫ x) ↔ (z ywa(b(b a ↔ (c(c bc = w) d(d b ↔ (d = w e(e de = z))))) a x)))
1716albii 1566 . . 3 (z(z yww, {z}⟫ x) ↔ z(z ywa(b(b a ↔ (c(c bc = w) d(d b ↔ (d = w e(e de = z))))) a x)))
1817exbii 1582 . 2 (yz(z yww, {z}⟫ x) ↔ yz(z ywa(b(b a ↔ (c(c bc = w) d(d b ↔ (d = w e(e de = z))))) a x)))
191, 18mpbi 199 1 yz(z ywa(b(b a ↔ (c(c bc = w) d(d b ↔ (d = w e(e de = z))))) a x))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {csn 3737  ⟪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-typlower 4086 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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