Theorem gencbval 1373

Description: Change of bound variable using implicit substitution.

Hypotheses

Ref	Expression
gencbval.1	⊢ A ∈ V
gencbval.2	⊢ (A = y → (φ ↔ ψ))
gencbval.3	⊢ (A = y → (χ ↔ θ))
gencbval.4	⊢ (θ ↔ ∃x(χ ∧ A = y))

Assertion

Ref	Expression
gencbval	⊢ (∀x(χ → φ) ↔ ∀y(θ → ψ))

Distinct variable group(s):   ψ,x   φ,y   θ,x   χ,y   y,A

Proof of Theorem gencbval

Step	Hyp	Ref	Expression
1		gencbval.1	. . . 4 ⊢ A ∈ V
2		gencbval.2	. . . . 5 ⊢ (A = y → (φ ↔ ψ))
3	2	negbid 463	. . . 4 ⊢ (A = y → (¬ φ ↔ ¬ ψ))
4		gencbval.3	. . . 4 ⊢ (A = y → (χ ↔ θ))
5		gencbval.4	. . . 4 ⊢ (θ ↔ ∃x(χ ∧ A = y))
6	1, 3, 4, 5	gencbvex 1372	. . 3 ⊢ (∃x(χ ∧ ¬ φ) ↔ ∃y(θ ∧ ¬ ψ))
7		exanali 725	. . 3 ⊢ (∃x(χ ∧ ¬ φ) ↔ ¬ ∀x(χ → φ))
8		exanali 725	. . 3 ⊢ (∃y(θ ∧ ¬ ψ) ↔ ¬ ∀y(θ → ψ))
9	6, 7, 8	3bitr3 156	. 2 ⊢ (¬ ∀x(χ → φ) ↔ ¬ ∀y(θ → ψ))
10	9	bicon4i 401	1 ⊢ (∀x(χ → φ) ↔ ∀y(θ → ψ))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  suppsr 4016  supsrlem6 4024  supre 4054

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-9 799  ax-12 802  ax-17 925  ax-ext 1074

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

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