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Theorem cbvralsv 3394
 Description: Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
cbvralsv (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem cbvralsv
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 2015 . . 3 𝑧𝜑
2 nfs1v 2313 . . 3 𝑥[𝑧 / 𝑥]𝜑
3 sbequ12 2288 . . 3 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
41, 2, 3cbvral 3379 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑧𝐴 [𝑧 / 𝑥]𝜑)
5 nfv 2015 . . . 4 𝑦𝜑
65nfsb 2575 . . 3 𝑦[𝑧 / 𝑥]𝜑
7 nfv 2015 . . 3 𝑧[𝑦 / 𝑥]𝜑
8 sbequ 2507 . . 3 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
96, 7, 8cbvral 3379 . 2 (∀𝑧𝐴 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
104, 9bitri 267 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198  [wsb 2069  ∀wral 3117 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clel 2821  df-nfc 2958  df-ral 3122 This theorem is referenced by:  sbralie  3396  rspsbc  3742  ralxpf  5501  tfinds  7320  tfindes  7323  nn0min  30114
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