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Theorem sbievw 2103
Description: Conversion of implicit substitution to explicit substitution. Version of sbie 2544 and sbiev 2330 with more disjoint variable conditions, requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by BJ, 18-Jul-2023.)
Hypothesis
Ref Expression
sbievw.is (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
sbievw ([𝑦 / 𝑥]𝜑𝜓)
Distinct variable groups:   𝑥,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem sbievw
StepHypRef Expression
1 sb6 2093 . 2 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
2 sbievw.is . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
32equsalvw 2010 . 2 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
41, 3bitri 277 1 ([𝑦 / 𝑥]𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wal 1535  [wsb 2069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070
This theorem is referenced by:  sbiedvw  2104  2sbievw  2105  sbievw2  2107  sbid2vw  2260  2mos  2733  cbvabv  2888  clelsb3vOLD  2939  sbralie  3450  elabgw  3644  sbcco2  3779  sbcie2g  3791  2reu8i  43460  ichcircshi  43762
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