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Theorem spsbbi 2069
Description: Biconditional property for substitution. Closed form of sbbii 2072. Specialization of biconditional. (Contributed by NM, 2-Jun-1993.) Revise df-sb 2061. (Revised by BJ, 22-Dec-2020.)
Assertion
Ref Expression
spsbbi (∀𝑥(𝜑𝜓) → ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓))

Proof of Theorem spsbbi
StepHypRef Expression
1 biimp 216 . . . 4 ((𝜑𝜓) → (𝜑𝜓))
21alimi 1803 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑥(𝜑𝜓))
3 spsbim 2068 . . 3 (∀𝑥(𝜑𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))
42, 3syl 17 . 2 (∀𝑥(𝜑𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))
5 biimpr 221 . . . 4 ((𝜑𝜓) → (𝜓𝜑))
65alimi 1803 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑥(𝜓𝜑))
7 spsbim 2068 . . 3 (∀𝑥(𝜓𝜑) → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜑))
86, 7syl 17 . 2 (∀𝑥(𝜑𝜓) → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜑))
94, 8impbid 213 1 (∀𝑥(𝜑𝜓) → ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1526  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902
This theorem depends on definitions:  df-bi 208  df-sb 2061
This theorem is referenced by:  sbbidv  2075  sbbid  2237  sbbibOLD  2281  abbi1  2884  sbeqi  35320
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