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Theorem cbvsbv 2098
Description: Change the bound variable (i.e. the substituted one) in wff's linked by implicit substitution. The proof was extracted from a former cbvabv 2810 version. (Contributed by Wolf Lammen, 16-Mar-2025.)
Hypothesis
Ref Expression
cbvsbv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvsbv ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑧)   𝜓(𝑦,𝑧)

Proof of Theorem cbvsbv
StepHypRef Expression
1 sbco2vv 2097 . 2 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
2 cbvsbv.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
32sbievw 2091 . . 3 ([𝑦 / 𝑥]𝜑𝜓)
43sbbii 2074 . 2 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
51, 4bitr3i 277 1 ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  [wsb 2062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063
This theorem is referenced by:  sbco4lem  2099  cbvabv  2810
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