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Theorem 2sbiev 2527
Description: Conversion of double implicit substitution to explicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2382. See 2sbievw 2103 for a version with extra disjoint variables, but based on fewer axioms. (Contributed by AV, 29-Jul-2023.) (New usage is discouraged.)
Hypothesis
Ref Expression
2sbiev.1 ((𝑥 = 𝑡𝑦 = 𝑢) → (𝜑𝜓))
Assertion
Ref Expression
2sbiev ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑𝜓)
Distinct variable groups:   𝑥,𝑦,𝜓   𝑦,𝑡
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑢,𝑡)   𝜓(𝑢,𝑡)

Proof of Theorem 2sbiev
StepHypRef Expression
1 nfv 1915 . 2 𝑥𝜓
2 2sbiev.1 . . 3 ((𝑥 = 𝑡𝑦 = 𝑢) → (𝜑𝜓))
32sbiedv 2526 . 2 (𝑥 = 𝑡 → ([𝑢 / 𝑦]𝜑𝜓))
41, 3sbie 2524 1 ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  [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  ax-6 1970  ax-7 2015  ax-10 2143  ax-12 2176  ax-13 2382
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070
This theorem is referenced by: (None)
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