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Theorem spsbbi 2078
 Description: Biconditional property for substitution. Closed form of sbbii 2081. Specialization of biconditional. (Contributed by NM, 2-Jun-1993.) Revise df-sb 2070. (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 218 . . . 4 ((𝜑𝜓) → (𝜑𝜓))
21alimi 1813 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑥(𝜑𝜓))
3 spsbim 2077 . . 3 (∀𝑥(𝜑𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))
42, 3syl 17 . 2 (∀𝑥(𝜑𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))
5 biimpr 223 . . . 4 ((𝜑𝜓) → (𝜓𝜑))
65alimi 1813 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑥(𝜓𝜑))
7 spsbim 2077 . . 3 (∀𝑥(𝜓𝜑) → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜑))
86, 7syl 17 . 2 (∀𝑥(𝜑𝜓) → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜑))
94, 8impbid 215 1 (∀𝑥(𝜑𝜓) → ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911 This theorem depends on definitions:  df-bi 210  df-sb 2070 This theorem is referenced by:  sbbidv  2084  sbbid  2244  sbbibOLD  2285  abbi1  2861  sbeqi  35616
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