Theorem gencbval 2904

Description: Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.)

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 groups:   ψ,x   φ,y   θ,x   χ,y   y,A

Allowed substitution hints:   φ(x)   ψ(y)   χ(x)   θ(y)   A(x)

Proof of Theorem gencbval

Step	Hyp	Ref	Expression
1		gencbval.1	. . . 4 ⊢ A ∈ V
2		gencbval.2	. . . . 5 ⊢ (A = y → (φ ↔ ψ))
3	2	notbid 285	. . . 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 2902	. . 3 ⊢ (∃x(χ ∧ ¬ φ) ↔ ∃y(θ ∧ ¬ ψ))
7		exanali 1585	. . 3 ⊢ (∃x(χ ∧ ¬ φ) ↔ ¬ ∀x(χ → φ))
8		exanali 1585	. . 3 ⊢ (∃y(θ ∧ ¬ ψ) ↔ ¬ ∀y(θ → ψ))
9	6, 7, 8	3bitr3i 266	. 2 ⊢ (¬ ∀x(χ → φ) ↔ ¬ ∀y(θ → ψ))
10	9	con4bii 288	1 ⊢ (∀x(χ → φ) ↔ ∀y(θ → ψ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860

This theorem was proved from 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-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by: (None)