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Theorem 2sbiev 2503
Description: Conversion of double implicit substitution to explicit substitution. (Contributed by AV, 29-Jul-2023.)
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 1896 . 2 𝑥𝜓
2 2sbiev.1 . . 3 ((𝑥 = 𝑡𝑦 = 𝑢) → (𝜑𝜓))
32sbiedv 2502 . 2 (𝑥 = 𝑡 → ([𝑢 / 𝑦]𝜑𝜓))
41, 3sbie 2500 1 ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  [wsb 2044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-10 2114  ax-12 2143  ax-13 2346
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1766  df-nf 1770  df-sb 2045
This theorem is referenced by:  ichbi12i  43122
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