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Theorem axcnvprim 3193
Description: ax-cnv 3182 presented without any set theory definitions.
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 3182 . 2 yzw(⟪z, w y ↔ ⟪w, z x)
2 df-clel 1891 . . . . . 6 (⟪z, w ya(a = ⟪z, w a y))
3 axprimlem2 3191 . . . . . . . 8 (a = ⟪z, w⟫ ↔ b(b a ↔ (c(c bc = z) d(d b ↔ (d = z d = w)))))
43anbi1i 665 . . . . . . 7 ((a = ⟪z, w a y) ↔ (b(b a ↔ (c(c bc = z) d(d b ↔ (d = z d = w)))) a y))
54exbii 1359 . . . . . 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 237 . . . . 5 (⟪z, w ya(b(b a ↔ (c(c bc = z) d(d b ↔ (d = z d = w)))) a y))
7 df-clel 1891 . . . . . 6 (⟪w, z xe(e = ⟪w, z e x))
8 axprimlem2 3191 . . . . . . . 8 (e = ⟪w, z⟫ ↔ f(f e ↔ (g(g fg = w) h(h f ↔ (h = w h = z)))))
98anbi1i 665 . . . . . . 7 ((e = ⟪w, z e x) ↔ (f(f e ↔ (g(g fg = w) h(h f ↔ (h = w h = z)))) e x))
109exbii 1359 . . . . . 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 237 . . . . 5 (⟪w, z xe(f(f e ↔ (g(g fg = w) h(h f ↔ (h = w h = z)))) e x))
126, 11bibi12i 303 . . . 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 1343 . . 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 1359 . 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 196 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 173   wo 354   wa 355  wal 1322  wex 1327   = wceq 1398   wcel 1400  copk 2862
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1323  ax-6 1324  ax-7 1325  ax-gen 1326  ax-8 1402  ax-10 1403  ax-11 1404  ax-12 1405  ax-17 1413  ax-9 1424  ax-4 1429  ax-16 1606  ax-ext 1877  ax-cnv 3182
This theorem depends on definitions:  df-bi 174  df-or 356  df-an 357  df-nand 1259  df-ex 1328  df-sb 1568  df-clab 1883  df-cleq 1888  df-clel 1891  df-v 2302  df-nin 2613  df-compl 2614  df-un 2616  df-sn 2807  df-pr 2808  df-opk 2863
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