Theorem axcnvprim 4092

Description: ax-cnv 4081 presented without any set theory definitions. (Contributed by SF, 25-Mar-2015.)

Assertion

Ref	Expression
axcnvprim	⊢ ∃y∀z∀w(∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = z) ∨ ∀d(d ∈ b ↔ (d = z ∨ d = w)))) ∧ a ∈ y) ↔ ∃e(∀f(f ∈ e ↔ (∀g(g ∈ f ↔ g = 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

Step	Hyp	Ref	Expression
1		ax-cnv 4081	. 2 ⊢ ∃y∀z∀w(⟪z, w⟫ ∈ y ↔ ⟪w, z⟫ ∈ x)
2		df-clel 2349	. . . . . 6 ⊢ (⟪z, w⟫ ∈ y ↔ ∃a(a = ⟪z, w⟫ ∧ a ∈ y))
3		axprimlem2 4090	. . . . . . . 8 ⊢ (a = ⟪z, w⟫ ↔ ∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = z) ∨ ∀d(d ∈ b ↔ (d = z ∨ d = w)))))
4	3	anbi1i 676	. . . . . . 7 ⊢ ((a = ⟪z, w⟫ ∧ a ∈ y) ↔ (∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = z) ∨ ∀d(d ∈ b ↔ (d = z ∨ d = w)))) ∧ a ∈ y))
5	4	exbii 1582	. . . . . 6 ⊢ (∃a(a = ⟪z, w⟫ ∧ a ∈ y) ↔ ∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = z) ∨ ∀d(d ∈ b ↔ (d = z ∨ d = w)))) ∧ a ∈ y))
6	2, 5	bitri 240	. . . . 5 ⊢ (⟪z, w⟫ ∈ y ↔ ∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = z) ∨ ∀d(d ∈ b ↔ (d = z ∨ d = w)))) ∧ a ∈ y))
7		df-clel 2349	. . . . . 6 ⊢ (⟪w, z⟫ ∈ x ↔ ∃e(e = ⟪w, z⟫ ∧ e ∈ x))
8		axprimlem2 4090	. . . . . . . 8 ⊢ (e = ⟪w, z⟫ ↔ ∀f(f ∈ e ↔ (∀g(g ∈ f ↔ g = w) ∨ ∀h(h ∈ f ↔ (h = w ∨ h = z)))))
9	8	anbi1i 676	. . . . . . 7 ⊢ ((e = ⟪w, z⟫ ∧ e ∈ x) ↔ (∀f(f ∈ e ↔ (∀g(g ∈ f ↔ g = w) ∨ ∀h(h ∈ f ↔ (h = w ∨ h = z)))) ∧ e ∈ x))
10	9	exbii 1582	. . . . . 6 ⊢ (∃e(e = ⟪w, z⟫ ∧ e ∈ x) ↔ ∃e(∀f(f ∈ e ↔ (∀g(g ∈ f ↔ g = w) ∨ ∀h(h ∈ f ↔ (h = w ∨ h = z)))) ∧ e ∈ x))
11	7, 10	bitri 240	. . . . 5 ⊢ (⟪w, z⟫ ∈ x ↔ ∃e(∀f(f ∈ e ↔ (∀g(g ∈ f ↔ g = w) ∨ ∀h(h ∈ f ↔ (h = w ∨ h = z)))) ∧ e ∈ x))
12	6, 11	bibi12i 306	. . . 4 ⊢ ((⟪z, w⟫ ∈ y ↔ ⟪w, z⟫ ∈ x) ↔ (∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = z) ∨ ∀d(d ∈ b ↔ (d = z ∨ d = w)))) ∧ a ∈ y) ↔ ∃e(∀f(f ∈ e ↔ (∀g(g ∈ f ↔ g = w) ∨ ∀h(h ∈ f ↔ (h = w ∨ h = z)))) ∧ e ∈ x)))
13	12	2albii 1567	. . 3 ⊢ (∀z∀w(⟪z, w⟫ ∈ y ↔ ⟪w, z⟫ ∈ x) ↔ ∀z∀w(∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = z) ∨ ∀d(d ∈ b ↔ (d = z ∨ d = w)))) ∧ a ∈ y) ↔ ∃e(∀f(f ∈ e ↔ (∀g(g ∈ f ↔ g = w) ∨ ∀h(h ∈ f ↔ (h = w ∨ h = z)))) ∧ e ∈ x)))
14	13	exbii 1582	. 2 ⊢ (∃y∀z∀w(⟪z, w⟫ ∈ y ↔ ⟪w, z⟫ ∈ x) ↔ ∃y∀z∀w(∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = z) ∨ ∀d(d ∈ b ↔ (d = z ∨ d = w)))) ∧ a ∈ y) ↔ ∃e(∀f(f ∈ e ↔ (∀g(g ∈ f ↔ g = w) ∨ ∀h(h ∈ f ↔ (h = w ∨ h = z)))) ∧ e ∈ x)))
15	1, 14	mpbi 199	1 ⊢ ∃y∀z∀w(∃a(∀b(b ∈ a ↔ (∀c(c ∈ b ↔ c = z) ∨ ∀d(d ∈ b ↔ (d = z ∨ d = w)))) ∧ a ∈ y) ↔ ∃e(∀f(f ∈ e ↔ (∀g(g ∈ f ↔ g = w) ∨ ∀h(h ∈ f ↔ (h = w ∨ h = z)))) ∧ e ∈ x))

This proof depends on syntax axioms:   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ⟪copk 4058

This proof depends on 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-cnv 4081

This proof 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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-opk 4059

This theorem is used by: (None)