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Theorem cbvralsvw 3315
Description: Change bound variable by using a substitution. Version of cbvralsv 3364 with a disjoint variable condition, which does not require ax-13 2375. (Contributed by NM, 20-Nov-2005.) Avoid ax-13 2375. (Revised by GG, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 8-Mar-2025.) Avoid ax-10 2139, ax-12 2175. (Revised by SN, 21-Aug-2025.)
Assertion
Ref Expression
cbvralsvw (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem cbvralsvw
StepHypRef Expression
1 sb8v 2353 . 2 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑦[𝑦 / 𝑥](𝑥𝐴𝜑))
2 df-ral 3060 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
3 df-ral 3060 . . 3 (∀𝑦𝐴 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦(𝑦𝐴 → [𝑦 / 𝑥]𝜑))
4 eleq1w 2822 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
54imbi1d 341 . . . . . 6 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜑)))
65sbbiiev 2090 . . . . 5 ([𝑦 / 𝑥](𝑥𝐴𝜑) ↔ [𝑦 / 𝑥](𝑦𝐴𝜑))
7 sbrimvw 2089 . . . . 5 ([𝑦 / 𝑥](𝑦𝐴𝜑) ↔ (𝑦𝐴 → [𝑦 / 𝑥]𝜑))
86, 7bitr2i 276 . . . 4 ((𝑦𝐴 → [𝑦 / 𝑥]𝜑) ↔ [𝑦 / 𝑥](𝑥𝐴𝜑))
98albii 1816 . . 3 (∀𝑦(𝑦𝐴 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑦[𝑦 / 𝑥](𝑥𝐴𝜑))
103, 9bitri 275 . 2 (∀𝑦𝐴 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦[𝑦 / 𝑥](𝑥𝐴𝜑))
111, 2, 103bitr4i 303 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535  [wsb 2062  wcel 2106  wral 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-11 2155
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clel 2814  df-ral 3060
This theorem is referenced by:  sbralieALT  3357  rspsbc  3888  ralxpf  5860  tfinds  7881  tfindes  7884  nn0min  32827
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