Theorem cbvabw 2801

Description: Rule used to change bound variables, using implicit substitution. Version of cbvab 2802 with a disjoint variable condition, which does not require ax-10 2142, ax-13 2371. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by GG, 23-May-2024.)

Hypotheses

Ref	Expression
cbvabw.1	⊢ Ⅎ𝑦𝜑
cbvabw.2	⊢ Ⅎ𝑥𝜓
cbvabw.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvabw	⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

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.

Step	Hyp	Ref	Expression
1		cbvabw.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2		cbvabw.2	. . . 4 ⊢ Ⅎ𝑥𝜓
3		cbvabw.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvsbvf 2362	. . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
5		df-clab 2709	. . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑)
6		df-clab 2709	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓)
7	4, 5, 6	3bitr4i 303	. 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜓})
8	7	eqriv 2727	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  Ⅎwnf 1783  [wsb 2065   ∈ wcel 2109  {cab 2708

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 2008  ax-9 2119  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722

This theorem is referenced by:  cbvrabw  3444  cbvrabwOLD  3445  cbvsbcw  3789  cbvrabcsfw  3906  rabsnifsb  4689  dfdmf  5863  dfrnf  5917  funfv2f  6953  abrexex2g  7946  bnj873  34921  fineqvrep  35092  ptrest  37620  poimirlem26  37647  poimirlem27  37648