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Theorem cbvralsv 3158
 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 1830 . . 3 𝑧𝜑
2 nfs1v 2425 . . 3 𝑥[𝑧 / 𝑥]𝜑
3 sbequ12 2097 . . 3 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
41, 2, 3cbvral 3143 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑧𝐴 [𝑧 / 𝑥]𝜑)
5 nfv 1830 . . . 4 𝑦𝜑
65nfsb 2428 . . 3 𝑦[𝑧 / 𝑥]𝜑
7 nfv 1830 . . 3 𝑧[𝑦 / 𝑥]𝜑
8 sbequ 2364 . . 3 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
96, 7, 8cbvral 3143 . 2 (∀𝑧𝐴 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
104, 9bitri 263 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  [wsb 1867  ∀wral 2896 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901 This theorem is referenced by:  sbralie  3160  rspsbc  3484  ralxpf  5190  tfinds  6951  tfindes  6954  nn0min  28954
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