Theorem hbcnv 2516

Description: Bound-variable hypothesis builder for converse.

Hypothesis

Ref	Expression
hbcnv.1	⊢ (y ∈ A → ∀x y ∈ A)

Assertion

Ref	Expression
hbcnv	⊢ (y ∈ ◡A → ∀x y ∈ ◡A)

Distinct variable group(s):   y,A   x,y

Proof of Theorem hbcnv

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . 5 ⊢ (y = ⟨z, w⟩ → ∀x y = ⟨z, w⟩)
2		ax-17 925	. . . . . 6 ⊢ (y ∈ w → ∀x y ∈ w)
3		hbcnv.1	. . . . . 6 ⊢ (y ∈ A → ∀x y ∈ A)
4		ax-17 925	. . . . . 6 ⊢ (y ∈ z → ∀x y ∈ z)
5	2, 3, 4	hbbr 2095	. . . . 5 ⊢ (wAz → ∀x wAz)
6	1, 5	hban 704	. . . 4 ⊢ ((y = ⟨z, w⟩ ∧ wAz) → ∀x(y = ⟨z, w⟩ ∧ wAz))
7	6	hbex 701	. . 3 ⊢ (∃w(y = ⟨z, w⟩ ∧ wAz) → ∀x∃w(y = ⟨z, w⟩ ∧ wAz))
8	7	hbex 701	. 2 ⊢ (∃z∃w(y = ⟨z, w⟩ ∧ wAz) → ∀x∃z∃w(y = ⟨z, w⟩ ∧ wAz))
9		elcnv 2514	. 2 ⊢ (y ∈ ◡A ↔ ∃z∃w(y = ⟨z, w⟩ ∧ wAz))
10	9	bial 695	. 2 ⊢ (∀x y ∈ ◡A ↔ ∀x∃z∃w(y = ⟨z, w⟩ ∧ wAz))
11	8, 9, 10	3imtr4 192	1 ⊢ (y ∈ ◡A → ∀x y ∈ ◡A)

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ⟨cop 1810   class class class wbr 2054  ^◡ccnv 2409

This theorem is referenced by:  hbdm 2565  hbfun 2684  hbf1 2779

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-cnv 2426

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