Theorem cbvabdavw 37139

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 37137	. . 3 ⊢ (𝜑 → ([𝑡 / 𝑥]𝜓 ↔ [𝑡 / 𝑦]𝜒))
3		df-clab 2739	. . 3 ⊢ (𝑡 ∈ {𝑥 ∣ 𝜓} ↔ [𝑡 / 𝑥]𝜓)
4		df-clab 2739	. . 3 ⊢ (𝑡 ∈ {𝑦 ∣ 𝜒} ↔ [𝑡 / 𝑦]𝜒)
5	2, 3, 4	3bitr4g 317	. 2 ⊢ (𝜑 → (𝑡 ∈ {𝑥 ∣ 𝜓} ↔ 𝑡 ∈ {𝑦 ∣ 𝜒}))
6	5	eqrdv 2758	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑦 ∣ 𝜒})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  [wsb 2098   ∈ wcel 2144  {cab 2738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752

This theorem is used by:  cbvsbcdavw  37140  cbvsbcdavw2  37141  cbvrabdavw  37144  cbviotadavw  37152  cbvixpdavw  37161  cbvrabdavw2  37168  cbvixpdavw2  37177