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Theorem bj-alsb 32302
Description: If a proposition is true for all instances, then it is true for any specific one. Uses only ax-1--5. Compare stdpc4 2352 which uses auxiliary axioms. (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 1738 . . . 4 (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
21a1d 25 . . 3 (∀𝑥𝜑 → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
32alrimiv 1852 . 2 (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
4 df-ssb 32297 . 2 ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
53, 4sylibr 224 1 (∀𝑥𝜑 → [𝑡/𝑥]b𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478  [wssb 32296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836
This theorem depends on definitions:  df-bi 197  df-ssb 32297
This theorem is referenced by:  bj-ssbft  32319
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