Theorem sbhb 2520

Description: Two ways of expressing "𝑥 is (effectively) not free in 𝜑". Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 29-May-2009.) (New usage is discouraged.)

Assertion

Ref	Expression
sbhb	⊢ ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))

Distinct variable group:   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sbhb

Step	Hyp	Ref	Expression
1		nfv 1917	. . . 4 ⊢ Ⅎ𝑦𝜑
2	1	sb8 2516	. . 3 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
3	2	imbi2i 335	. 2 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑))
4		19.21v 1942	. 2 ⊢ (∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑) ↔ (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑))
5	3, 4	bitr4i 277	1 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1539  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2371

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-nf 1786  df-sb 2068

This theorem is referenced by: (None)