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Theorem bj-alsb 31814
 Description: If a proposition is true for all instances, then it is true for any specific one. See stdpc4 2341. (Contributed by BJ, 22-Dec-2020.)
Assertion
Ref Expression
bj-alsb (∀𝑥𝜑 → [𝑡/𝑥]b𝜑)

Proof of Theorem bj-alsb
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ala1 1755 . . . 4 (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
21a1d 25 . . 3 (∀𝑥𝜑 → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
32alrimiv 1842 . 2 (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
4 df-ssb 31809 . 2 ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
53, 4sylibr 223 1 (∀𝑥𝜑 → [𝑡/𝑥]b𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473  [wssb 31808 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827 This theorem depends on definitions:  df-bi 196  df-ssb 31809 This theorem is referenced by:  bj-ssbft  31831
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