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Theorem cbvralsvw 3379
 Description: Change bound variable by using a substitution. Version of cbvralsv 3381 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 20-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.)
Assertion
Ref Expression
cbvralsvw (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
Distinct variable groups:   𝑥,𝐴,𝑦   𝜑,𝑦   𝑥,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem cbvralsvw
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1915 . . 3 𝑧𝜑
2 nfs1v 2157 . . 3 𝑥[𝑧 / 𝑥]𝜑
3 sbequ12 2250 . . 3 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
41, 2, 3cbvralw 3352 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑧𝐴 [𝑧 / 𝑥]𝜑)
5 nfv 1915 . . 3 𝑦[𝑧 / 𝑥]𝜑
6 nfv 1915 . . 3 𝑧[𝑦 / 𝑥]𝜑
7 sbequ 2088 . . 3 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
85, 6, 7cbvralw 3352 . 2 (∀𝑧𝐴 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
94, 8bitri 278 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  [wsb 2069  ∀wral 3070 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-10 2142  ax-11 2158  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clel 2830  df-nfc 2901  df-ral 3075 This theorem is referenced by:  sbralie  3383  rspsbc  3785  ralxpf  5686  tfinds  7573  tfindes  7576  nn0min  30658
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