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Theorem 2sbbid 2248
Description: Deduction doubly substituting both sides of a biconditional. (Contributed by AV, 30-Jul-2023.)
Hypotheses
Ref Expression
sbbid.1 𝑥𝜑
sbbid.2 (𝜑 → (𝜓𝜒))
2sbbid.1 𝑦𝜑
Assertion
Ref Expression
2sbbid (𝜑 → ([𝑡 / 𝑥][𝑢 / 𝑦]𝜓 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜒))

Proof of Theorem 2sbbid
StepHypRef Expression
1 sbbid.1 . 2 𝑥𝜑
2 2sbbid.1 . . 3 𝑦𝜑
3 sbbid.2 . . 3 (𝜑 → (𝜓𝜒))
42, 3sbbid 2247 . 2 (𝜑 → ([𝑢 / 𝑦]𝜓 ↔ [𝑢 / 𝑦]𝜒))
51, 4sbbid 2247 1 (𝜑 → ([𝑡 / 𝑥][𝑢 / 𝑦]𝜓 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wnf 1785  [wsb 2070
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 1912  ax-6 1971  ax-7 2016  ax-12 2178
This theorem depends on definitions:  df-bi 210  df-ex 1782  df-nf 1786  df-sb 2071
This theorem is referenced by: (None)
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