Theorem wl-dfralsb 35002

Description: An alternate definition of restricted universal quantification (df-wl-ral 35001) using substitution. (Contributed by Wolf Lammen, 25-May-2023.)

Assertion

Ref	Expression
wl-dfralsb	⊢ (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → [𝑦 / 𝑥]𝜑))

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem wl-dfralsb

Step	Hyp	Ref	Expression
1		df-wl-ral 35001	. 2 ⊢ (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		sb6 2090	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	2	imbi2i 339	. . 3 ⊢ ((𝑦 ∈ 𝐴 → [𝑦 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
4	3	albii 1821	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
5	1, 4	bitr4i 281	1 ⊢ (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → [𝑦 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  [wsb 2069   ∈ wcel 2111  ∀wl-ral 34996

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 1911  ax-6 1970  ax-7 2015

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-wl-ral 35001

This theorem is referenced by:  wl-rgenw  35008  wl-dfrexsb  35016  wl-dfrmosb  35018