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Theorem cbvralsv 3177
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 1841 . . 3 𝑧𝜑
2 nfs1v 2435 . . 3 𝑥[𝑧 / 𝑥]𝜑
3 sbequ12 2109 . . 3 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
41, 2, 3cbvral 3162 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑧𝐴 [𝑧 / 𝑥]𝜑)
5 nfv 1841 . . . 4 𝑦𝜑
65nfsb 2438 . . 3 𝑦[𝑧 / 𝑥]𝜑
7 nfv 1841 . . 3 𝑧[𝑦 / 𝑥]𝜑
8 sbequ 2374 . . 3 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
96, 7, 8cbvral 3162 . 2 (∀𝑧𝐴 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
104, 9bitri 264 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  [wsb 1878  wral 2909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914
This theorem is referenced by:  sbralie  3179  rspsbc  3511  ralxpf  5257  tfinds  7044  tfindes  7047  nn0min  29541
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