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Theorem cbvralsv 3154
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 1829 . . 3 𝑧𝜑
2 nfs1v 2421 . . 3 𝑥[𝑧 / 𝑥]𝜑
3 sbequ12 2095 . . 3 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
41, 2, 3cbvral 3139 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑧𝐴 [𝑧 / 𝑥]𝜑)
5 nfv 1829 . . . 4 𝑦𝜑
65nfsb 2424 . . 3 𝑦[𝑧 / 𝑥]𝜑
7 nfv 1829 . . 3 𝑧[𝑦 / 𝑥]𝜑
8 sbequ 2360 . . 3 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
96, 7, 8cbvral 3139 . 2 (∀𝑧𝐴 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
104, 9bitri 262 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 194  [wsb 1866  wral 2892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897
This theorem is referenced by:  sbralie  3156  rspsbc  3480  ralxpf  5175  tfinds  6925  tfindes  6928  nn0min  28757
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