Theorem wl-dfralsb 34852

Description: An alternate definition of restricted universal quantification (df-wl-ral 34851) 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 34851	. 2 ⊢ (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		sb6 2093	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	2	imbi2i 338	. . 3 ⊢ ((𝑦 ∈ 𝐴 → [𝑦 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
4	3	albii 1820	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
5	1, 4	bitr4i 280	1 ⊢ (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → [𝑦 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535  [wsb 2069   ∈ wcel 2114  ∀wl-ral 34846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-wl-ral 34851

This theorem is referenced by:  wl-rgenw  34858  wl-dfrexsb  34866  wl-dfrmosb  34868