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Theorem sb2ae 2502
Description: In the case of two successive substitutions for two always equal variables, the second substitution has no effect. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by BJ and WL, 9-Aug-2023.) (New usage is discouraged.)
Assertion
Ref Expression
sb2ae (∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑))
Distinct variable group:   𝑦,𝑣
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑣,𝑢)

Proof of Theorem sb2ae
StepHypRef Expression
1 drsb1 2501 . 2 (∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑢 / 𝑦][𝑣 / 𝑦]𝜑))
2 nfs1v 2157 . . 3 𝑦[𝑣 / 𝑦]𝜑
32sbf 2267 . 2 ([𝑢 / 𝑦][𝑣 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑)
41, 3bitrdi 287 1 (∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1540  [wsb 2071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-10 2141  ax-12 2175  ax-13 2374
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1787  df-nf 1791  df-sb 2072
This theorem is referenced by: (None)
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