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Theorem cbvabw 2811
Description: Rule used to change bound variables, using implicit substitution. Version of cbvab 2812 with a disjoint variable condition, which does not require ax-10 2139, ax-13 2375. (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 2364 . . 3 ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
5 df-clab 2713 . . 3 (𝑧 ∈ {𝑥𝜑} ↔ [𝑧 / 𝑥]𝜑)
6 df-clab 2713 . . 3 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
74, 5, 63bitr4i 303 . 2 (𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦𝜓})
87eqriv 2732 1 {𝑥𝜑} = {𝑦𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wnf 1780  [wsb 2062  wcel 2106  {cab 2712
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-9 2116  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727
This theorem is referenced by:  cbvrabw  3471  cbvrabwOLD  3472  cbvsbcw  3825  cbvrabcsfw  3952  rabsnifsb  4727  dfdmf  5910  dfrnf  5964  funfv2f  6998  abrexex2g  7988  bnj873  34917  fineqvrep  35088  ptrest  37606  poimirlem26  37633  poimirlem27  37634
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