Theorem cbvabw 2840

Description: Rule used to change bound variables, using implicit substitution. Version of cbvab 2841 with a disjoint variable condition, which does not require ax-10 2182, ax-13 2410. (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 2401	. . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
5		df-clab 2748	. . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑)
6		df-clab 2748	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓)
7	4, 5, 6	3bitr4i 306	. 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜓})
8	7	eqriv 2766	1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567  Ⅎwnf 1810  [wsb 2097   ∈ wcel 2149  {cab 2747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761

This theorem is referenced by:  cbvrabw  3457  cbvsbcw  3784  cbvrabcsfw  3900  rabsnifsb  4691  dfdmf  5887  dfrnf  5941  funfv2f  6971  abrexex2g  7961  bnj873  35257  fineqvrep  35460  ptrest  38193  poimirlem26  38220  poimirlem27  38221