Theorem hbsb4t 906

Description: A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 905).

Assertion

Ref	Expression
hbsb4t	⊢ (∀x∀z(φ → ∀zφ) → (¬ ∀z z = y → ([y / x]φ → ∀z[y / x]φ)))

Proof of Theorem hbsb4t

Step	Hyp	Ref	Expression
1		ax-4 673	. . . . . 6 ⊢ (∀zφ → φ)
2	1	biantru 543	. . . . 5 ⊢ ((φ → ∀zφ) ↔ ((φ → ∀zφ) ∧ (∀zφ → φ)))
3		bi 396	. . . . 5 ⊢ ((φ ↔ ∀zφ) ↔ ((φ → ∀zφ) ∧ (∀zφ → φ)))
4	2, 3	bitr4 154	. . . 4 ⊢ ((φ → ∀zφ) ↔ (φ ↔ ∀zφ))
5	4	bi2al 696	. . 3 ⊢ (∀x∀z(φ → ∀zφ) ↔ ∀x∀z(φ ↔ ∀zφ))
6		sbba4 896	. . . . . 6 ⊢ (∀x(φ ↔ ∀zφ) → ([y / x]φ ↔ [y / x]∀zφ))
7	6	a4s 682	. . . . 5 ⊢ (∀z∀x(φ ↔ ∀zφ) → ([y / x]φ ↔ [y / x]∀zφ))
8		hba1 698	. . . . . 6 ⊢ (∀z∀x(φ ↔ ∀zφ) → ∀z∀z∀x(φ ↔ ∀zφ))
9	8, 7	biald 782	. . . . 5 ⊢ (∀z∀x(φ ↔ ∀zφ) → (∀z[y / x]φ ↔ ∀z[y / x]∀zφ))
10	7, 9	imbi12d 474	. . . 4 ⊢ (∀z∀x(φ ↔ ∀zφ) → (([y / x]φ → ∀z[y / x]φ) ↔ ([y / x]∀zφ → ∀z[y / x]∀zφ)))
11	10	a7s 689	. . 3 ⊢ (∀x∀z(φ ↔ ∀zφ) → (([y / x]φ → ∀z[y / x]φ) ↔ ([y / x]∀zφ → ∀z[y / x]∀zφ)))
12	5, 11	sylbi 174	. 2 ⊢ (∀x∀z(φ → ∀zφ) → (([y / x]φ → ∀z[y / x]φ) ↔ ([y / x]∀zφ → ∀z[y / x]∀zφ)))
13		hba1 698	. . 3 ⊢ (∀zφ → ∀z∀zφ)
14	13	hbsb4 905	. 2 ⊢ (¬ ∀z z = y → ([y / x]∀zφ → ∀z[y / x]∀zφ))
15	12, 14	syl5bir 184	1 ⊢ (∀x∀z(φ → ∀zφ) → (¬ ∀z z = y → ([y / x]φ → ∀z[y / x]φ)))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   = weq 797  [wsb 852

This theorem is referenced by:  ddelimdf 909

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

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853

metamath.org