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Theorem axcnvprim 4091
Description: ax-cnv 4080 presented without any set theory definitions. (Contributed by SF, 25-Mar-2015.)
Assertion
Ref Expression
axcnvprim yzw(a(b(b a ↔ (c(c bc = z) d(d b ↔ (d = z d = w)))) a y) ↔ e(f(f e ↔ (g(g fg = w) h(h f ↔ (h = w h = z)))) e x))
Distinct variable groups:   a,b,w   y,a,z   b,c   b,d,w,z   z,c   w,d,z   e,f,w   x,e,z   f,g   f,h,w,z   w,g   w,h,z   x,w,y,z

Proof of Theorem axcnvprim
StepHypRef Expression
1 ax-cnv 4080 . 2 yzw(⟪z, w y ↔ ⟪w, z x)
2 df-clel 2349 . . . . . 6 (⟪z, w ya(a = ⟪z, w a y))
3 axprimlem2 4089 . . . . . . . 8 (a = ⟪z, w⟫ ↔ b(b a ↔ (c(c bc = z) d(d b ↔ (d = z d = w)))))
43anbi1i 676 . . . . . . 7 ((a = ⟪z, w a y) ↔ (b(b a ↔ (c(c bc = z) d(d b ↔ (d = z d = w)))) a y))
54exbii 1582 . . . . . 6 (a(a = ⟪z, w a y) ↔ a(b(b a ↔ (c(c bc = z) d(d b ↔ (d = z d = w)))) a y))
62, 5bitri 240 . . . . 5 (⟪z, w ya(b(b a ↔ (c(c bc = z) d(d b ↔ (d = z d = w)))) a y))
7 df-clel 2349 . . . . . 6 (⟪w, z xe(e = ⟪w, z e x))
8 axprimlem2 4089 . . . . . . . 8 (e = ⟪w, z⟫ ↔ f(f e ↔ (g(g fg = w) h(h f ↔ (h = w h = z)))))
98anbi1i 676 . . . . . . 7 ((e = ⟪w, z e x) ↔ (f(f e ↔ (g(g fg = w) h(h f ↔ (h = w h = z)))) e x))
109exbii 1582 . . . . . 6 (e(e = ⟪w, z e x) ↔ e(f(f e ↔ (g(g fg = w) h(h f ↔ (h = w h = z)))) e x))
117, 10bitri 240 . . . . 5 (⟪w, z xe(f(f e ↔ (g(g fg = w) h(h f ↔ (h = w h = z)))) e x))
126, 11bibi12i 306 . . . 4 ((⟪z, w y ↔ ⟪w, z x) ↔ (a(b(b a ↔ (c(c bc = z) d(d b ↔ (d = z d = w)))) a y) ↔ e(f(f e ↔ (g(g fg = w) h(h f ↔ (h = w h = z)))) e x)))
13122albii 1567 . . 3 (zw(⟪z, w y ↔ ⟪w, z x) ↔ zw(a(b(b a ↔ (c(c bc = z) d(d b ↔ (d = z d = w)))) a y) ↔ e(f(f e ↔ (g(g fg = w) h(h f ↔ (h = w h = z)))) e x)))
1413exbii 1582 . 2 (yzw(⟪z, w y ↔ ⟪w, z x) ↔ yzw(a(b(b a ↔ (c(c bc = z) d(d b ↔ (d = z d = w)))) a y) ↔ e(f(f e ↔ (g(g fg = w) h(h f ↔ (h = w h = z)))) e x)))
151, 14mpbi 199 1 yzw(a(b(b a ↔ (c(c bc = z) d(d b ↔ (d = z d = w)))) a y) ↔ e(f(f e ↔ (g(g fg = w) h(h f ↔ (h = w h = z)))) e x))
Colors of variables: wff set class
Syntax hints:  wb 176   wo 357   wa 358  wal 1540  wex 1541   = wceq 1642   wcel 1710  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-cnv 4080
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
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