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Theorem sbbid 2244
Description: Deduction substituting both sides of a biconditional. (Contributed by NM, 30-Jun-1993.) Remove dependency on ax-10 2139 and ax-13 2375. (Revised by Wolf Lammen, 24-Nov-2022.) Revise df-sb 2063. (Revised by Steven Nguyen, 11-Jul-2023.)
Hypotheses
Ref Expression
sbbid.1 𝑥𝜑
sbbid.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
sbbid (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))

Proof of Theorem sbbid
StepHypRef Expression
1 sbbid.1 . . 3 𝑥𝜑
2 sbbid.2 . . 3 (𝜑 → (𝜓𝜒))
31, 2alrimi 2211 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 spsbbi 2071 . 2 (∀𝑥(𝜓𝜒) → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))
53, 4syl 17 1 (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535  wnf 1780  [wsb 2062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-ex 1777  df-nf 1781  df-sb 2063
This theorem is referenced by:  2sbbid  2245  sbcom3  2509  sbco3  2516  wl-sbcom2d-lem1  37540  wl-2spsbbi  37546  wl-clabt  37579
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