Theorem sbhb 2588

Description: Two ways of expressing "𝑥 is (effectively) not free in 𝜑." (Contributed by NM, 29-May-2009.)

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 1995	. . . 4 ⊢ Ⅎ𝑦𝜑
2	1	sb8 2571	. . 3 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
3	2	imbi2i 325	. 2 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑))
4		19.21v 2020	. 2 ⊢ (∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑) ↔ (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑))
5	3, 4	bitr4i 267	1 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1629  [wsb 2049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ex 1853  df-nf 1858  df-sb 2050

This theorem is referenced by: (None)