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Theorem cbvabw 2813
Description: Rule used to change bound variables, using implicit substitution. Version of cbvab 2814 with a disjoint variable condition, which does not require ax-10 2141, ax-13 2377. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by GG, 23-May-2024.)
Hypotheses
Ref Expression
cbvabw.1 𝑦𝜑
cbvabw.2 𝑥𝜓
cbvabw.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvabw {𝑥𝜑} = {𝑦𝜓}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvabw
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbvabw.1 . . . 4 𝑦𝜑
2 cbvabw.2 . . . 4 𝑥𝜓
3 cbvabw.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvsbvf 2366 . . 3 ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
5 df-clab 2715 . . 3 (𝑧 ∈ {𝑥𝜑} ↔ [𝑧 / 𝑥]𝜑)
6 df-clab 2715 . . 3 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
74, 5, 63bitr4i 303 . 2 (𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦𝜓})
87eqriv 2734 1 {𝑥𝜑} = {𝑦𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wnf 1783  [wsb 2064  wcel 2108  {cab 2714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729
This theorem is referenced by:  cbvrabw  3473  cbvrabwOLD  3474  cbvsbcw  3821  cbvrabcsfw  3940  rabsnifsb  4722  dfdmf  5907  dfrnf  5961  funfv2f  6998  abrexex2g  7989  bnj873  34938  fineqvrep  35109  ptrest  37626  poimirlem26  37653  poimirlem27  37654
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