Theorem cbvabdavw 36195

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 36193	. . 3 ⊢ (𝜑 → ([𝑡 / 𝑥]𝜓 ↔ [𝑡 / 𝑦]𝜒))
3		df-clab 2713	. . 3 ⊢ (𝑡 ∈ {𝑥 ∣ 𝜓} ↔ [𝑡 / 𝑥]𝜓)
4		df-clab 2713	. . 3 ⊢ (𝑡 ∈ {𝑦 ∣ 𝜒} ↔ [𝑡 / 𝑦]𝜒)
5	2, 3, 4	3bitr4g 314	. 2 ⊢ (𝜑 → (𝑡 ∈ {𝑥 ∣ 𝜓} ↔ 𝑡 ∈ {𝑦 ∣ 𝜒}))
6	5	eqrdv 2732	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑦 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539  [wsb 2063   ∈ wcel 2107  {cab 2712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726

This theorem is referenced by:  cbvsbcdavw  36196  cbvsbcdavw2  36197  cbvrabdavw  36200  cbviotadavw  36208  cbvixpdavw  36217  cbvrabdavw2  36224  cbvixpdavw2  36233