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Theorem 2sbievw 2103
 Description: Conversion of double implicit substitution to explicit substitution. Version of 2sbiev 2525 with more disjoint variable conditions, requiring fewer axioms. (Contributed by AV, 29-Jul-2023.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypothesis
Ref Expression
2sbievw.1 ((𝑥 = 𝑡𝑦 = 𝑢) → (𝜑𝜓))
Assertion
Ref Expression
2sbievw ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑𝜓)
Distinct variable groups:   𝑥,𝑦,𝜓   𝑥,𝑡,𝑦   𝑦,𝑢
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑢,𝑡)   𝜓(𝑢,𝑡)

Proof of Theorem 2sbievw
StepHypRef Expression
1 2sbievw.1 . . 3 ((𝑥 = 𝑡𝑦 = 𝑢) → (𝜑𝜓))
21sbiedvw 2102 . 2 (𝑥 = 𝑡 → ([𝑢 / 𝑦]𝜑𝜓))
32sbievw 2101 1 ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  [wsb 2070 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016 This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1783  df-sb 2071 This theorem is referenced by:  prtlem5  36421  ichbi12i  44330
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