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Theorem cbvralsv 3475
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 1908 . . 3 𝑧𝜑
2 nfs1v 2267 . . 3 𝑥[𝑧 / 𝑥]𝜑
3 sbequ12 2246 . . 3 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
41, 2, 3cbvral 3451 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑧𝐴 [𝑧 / 𝑥]𝜑)
5 nfv 1908 . . . 4 𝑦𝜑
65nfsb 2563 . . 3 𝑦[𝑧 / 𝑥]𝜑
7 nfv 1908 . . 3 𝑧[𝑦 / 𝑥]𝜑
8 sbequ 2083 . . 3 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
96, 7, 8cbvral 3451 . 2 (∀𝑧𝐴 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
104, 9bitri 276 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 207  [wsb 2062  wral 3143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clel 2898  df-nfc 2968  df-ral 3148
This theorem is referenced by:  nn0min  30450
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