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Theorem wl-dfralsb 34707
Description: An alternate definition of restricted universal quantification (df-wl-ral 34706) using substitution. (Contributed by Wolf Lammen, 25-May-2023.)
Assertion
Ref Expression
wl-dfralsb (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑦(𝑦𝐴 → [𝑦 / 𝑥]𝜑))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem wl-dfralsb
StepHypRef Expression
1 df-wl-ral 34706 . 2 (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑦(𝑦𝐴 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 sb6 2086 . . . 4 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
32imbi2i 337 . . 3 ((𝑦𝐴 → [𝑦 / 𝑥]𝜑) ↔ (𝑦𝐴 → ∀𝑥(𝑥 = 𝑦𝜑)))
43albii 1813 . 2 (∀𝑦(𝑦𝐴 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑦(𝑦𝐴 → ∀𝑥(𝑥 = 𝑦𝜑)))
51, 4bitr4i 279 1 (∀(𝑥 : 𝐴)𝜑 ↔ ∀𝑦(𝑦𝐴 → [𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1528  [wsb 2062  wcel 2107  wl-ral 34701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-sb 2063  df-wl-ral 34706
This theorem is referenced by:  wl-rgenw  34713  wl-dfrexsb  34721  wl-dfrmosb  34723
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