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Theorem axsiprim 4093
 Description: ax-si 4083 presented without any set theory definitions. (Contributed by SF, 25-Mar-2015.)
Assertion
Ref Expression
axsiprim yzw(a(b(b a ↔ (c(c bd(d cd = z)) e(e b ↔ (f(f ef = z) g(g eg = w))))) a y) ↔ h(i(i h ↔ (j(j ij = z) k(k i ↔ (k = z k = w)))) h x))
Distinct variable groups:   a,b,w   y,a,z   b,c   e,b,w,z   c,d,z   e,f   e,g,w   z,e,f   w,g   h,i,w   x,h,z   i,j   i,k,w,z   z,j   w,k,z   x,w,y,z

Proof of Theorem axsiprim
StepHypRef Expression
1 ax-si 4083 . 2 yzw(⟪{z}, {w}⟫ y ↔ ⟪z, w x)
2 df-clel 2349 . . . . . 6 (⟪{z}, {w}⟫ ya(a = ⟪{z}, {w}⟫ a y))
3 axprimlem2 4089 . . . . . . . . 9 (a = ⟪{z}, {w}⟫ ↔ b(b a ↔ (c(c bc = {z}) e(e b ↔ (e = {z} e = {w})))))
4 axprimlem1 4088 . . . . . . . . . . . . . 14 (c = {z} ↔ d(d cd = z))
54bibi2i 304 . . . . . . . . . . . . 13 ((c bc = {z}) ↔ (c bd(d cd = z)))
65albii 1566 . . . . . . . . . . . 12 (c(c bc = {z}) ↔ c(c bd(d cd = z)))
7 axprimlem1 4088 . . . . . . . . . . . . . . 15 (e = {z} ↔ f(f ef = z))
8 axprimlem1 4088 . . . . . . . . . . . . . . 15 (e = {w} ↔ g(g eg = w))
97, 8orbi12i 507 . . . . . . . . . . . . . 14 ((e = {z} e = {w}) ↔ (f(f ef = z) g(g eg = w)))
109bibi2i 304 . . . . . . . . . . . . 13 ((e b ↔ (e = {z} e = {w})) ↔ (e b ↔ (f(f ef = z) g(g eg = w))))
1110albii 1566 . . . . . . . . . . . 12 (e(e b ↔ (e = {z} e = {w})) ↔ e(e b ↔ (f(f ef = z) g(g eg = w))))
126, 11orbi12i 507 . . . . . . . . . . 11 ((c(c bc = {z}) e(e b ↔ (e = {z} e = {w}))) ↔ (c(c bd(d cd = z)) e(e b ↔ (f(f ef = z) g(g eg = w)))))
1312bibi2i 304 . . . . . . . . . 10 ((b a ↔ (c(c bc = {z}) e(e b ↔ (e = {z} e = {w})))) ↔ (b a ↔ (c(c bd(d cd = z)) e(e b ↔ (f(f ef = z) g(g eg = w))))))
1413albii 1566 . . . . . . . . 9 (b(b a ↔ (c(c bc = {z}) e(e b ↔ (e = {z} e = {w})))) ↔ b(b a ↔ (c(c bd(d cd = z)) e(e b ↔ (f(f ef = z) g(g eg = w))))))
153, 14bitri 240 . . . . . . . 8 (a = ⟪{z}, {w}⟫ ↔ b(b a ↔ (c(c bd(d cd = z)) e(e b ↔ (f(f ef = z) g(g eg = w))))))
1615anbi1i 676 . . . . . . 7 ((a = ⟪{z}, {w}⟫ a y) ↔ (b(b a ↔ (c(c bd(d cd = z)) e(e b ↔ (f(f ef = z) g(g eg = w))))) a y))
1716exbii 1582 . . . . . 6 (a(a = ⟪{z}, {w}⟫ a y) ↔ a(b(b a ↔ (c(c bd(d cd = z)) e(e b ↔ (f(f ef = z) g(g eg = w))))) a y))
182, 17bitri 240 . . . . 5 (⟪{z}, {w}⟫ ya(b(b a ↔ (c(c bd(d cd = z)) e(e b ↔ (f(f ef = z) g(g eg = w))))) a y))
19 df-clel 2349 . . . . . 6 (⟪z, w xh(h = ⟪z, w h x))
20 axprimlem2 4089 . . . . . . . 8 (h = ⟪z, w⟫ ↔ i(i h ↔ (j(j ij = z) k(k i ↔ (k = z k = w)))))
2120anbi1i 676 . . . . . . 7 ((h = ⟪z, w h x) ↔ (i(i h ↔ (j(j ij = z) k(k i ↔ (k = z k = w)))) h x))
2221exbii 1582 . . . . . 6 (h(h = ⟪z, w h x) ↔ h(i(i h ↔ (j(j ij = z) k(k i ↔ (k = z k = w)))) h x))
2319, 22bitri 240 . . . . 5 (⟪z, w xh(i(i h ↔ (j(j ij = z) k(k i ↔ (k = z k = w)))) h x))
2418, 23bibi12i 306 . . . 4 ((⟪{z}, {w}⟫ y ↔ ⟪z, w x) ↔ (a(b(b a ↔ (c(c bd(d cd = z)) e(e b ↔ (f(f ef = z) g(g eg = w))))) a y) ↔ h(i(i h ↔ (j(j ij = z) k(k i ↔ (k = z k = w)))) h x)))
25242albii 1567 . . 3 (zw(⟪{z}, {w}⟫ y ↔ ⟪z, w x) ↔ zw(a(b(b a ↔ (c(c bd(d cd = z)) e(e b ↔ (f(f ef = z) g(g eg = w))))) a y) ↔ h(i(i h ↔ (j(j ij = z) k(k i ↔ (k = z k = w)))) h x)))
2625exbii 1582 . 2 (yzw(⟪{z}, {w}⟫ y ↔ ⟪z, w x) ↔ yzw(a(b(b a ↔ (c(c bd(d cd = z)) e(e b ↔ (f(f ef = z) g(g eg = w))))) a y) ↔ h(i(i h ↔ (j(j ij = z) k(k i ↔ (k = z k = w)))) h x)))
271, 26mpbi 199 1 yzw(a(b(b a ↔ (c(c bd(d cd = z)) e(e b ↔ (f(f ef = z) g(g eg = w))))) a y) ↔ h(i(i h ↔ (j(j ij = z) k(k i ↔ (k = z k = w)))) h 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-mp 5  ax-1 6  ax-2 7  ax-3 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-si 4083 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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