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Theorem cbvralsvw 3466
Description: Version of cbvralsv 3468 with a disjoint variable condition, which does not require ax-13 2383. (Contributed 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 1908 . . 3 𝑧𝜑
2 nfs1v 2266 . . 3 𝑥[𝑧 / 𝑥]𝜑
3 sbequ12 2245 . . 3 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
41, 2, 3cbvralw 3440 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑧𝐴 [𝑧 / 𝑥]𝜑)
5 nfv 1908 . . 3 𝑦[𝑧 / 𝑥]𝜑
6 nfv 1908 . . 3 𝑧[𝑦 / 𝑥]𝜑
7 sbequ 2083 . . 3 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
85, 6, 7cbvralw 3440 . 2 (∀𝑧𝐴 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
94, 8bitri 277 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 208  [wsb 2062  wral 3136
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
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1774  df-nf 1778  df-sb 2063  df-clel 2891  df-nfc 2961  df-ral 3141
This theorem is referenced by:  sbralie  3470  rspsbc  3860  ralxpf  5710  tfinds  7566  tfindes  7569  nn0min  30529
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