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Theorem sb2ae 2514
 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 2379. (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 2513 . 2 (∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑢 / 𝑦][𝑣 / 𝑦]𝜑))
2 nfs1v 2157 . . 3 𝑦[𝑣 / 𝑦]𝜑
32sbf 2268 . 2 ([𝑢 / 𝑦][𝑣 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑)
41, 3syl6bb 290 1 (∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175  ax-13 2379 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070 This theorem is referenced by: (None)
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