Theorem cbvabdavw 36450

Description: Change bound variable in class abstractions. Deduction form. (Contributed by GG, 14-Aug-2025.)

Hypothesis

Ref	Expression
cbvabdavw.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
cbvabdavw	⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑦 ∣ 𝜒})

Distinct variable groups:   𝜑,𝑥,𝑦   𝜓,𝑦   𝜒,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem cbvabdavw

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvabdavw.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
2	1	cbvsbdavw 36448	. . 3 ⊢ (𝜑 → ([𝑡 / 𝑥]𝜓 ↔ [𝑡 / 𝑦]𝜒))
3		df-clab 2715	. . 3 ⊢ (𝑡 ∈ {𝑥 ∣ 𝜓} ↔ [𝑡 / 𝑥]𝜓)
4		df-clab 2715	. . 3 ⊢ (𝑡 ∈ {𝑦 ∣ 𝜒} ↔ [𝑡 / 𝑦]𝜒)
5	2, 3, 4	3bitr4g 314	. 2 ⊢ (𝜑 → (𝑡 ∈ {𝑥 ∣ 𝜓} ↔ 𝑡 ∈ {𝑦 ∣ 𝜒}))
6	5	eqrdv 2734	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑦 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  [wsb 2067   ∈ wcel 2113  {cab 2714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728

This theorem is referenced by:  cbvsbcdavw  36451  cbvsbcdavw2  36452  cbvrabdavw  36455  cbviotadavw  36463  cbvixpdavw  36472  cbvrabdavw2  36479  cbvixpdavw2  36488