Theorem cbvriotavw 7242

Description: Change bound variable in a restricted description binder. Version of cbvriotav 7247 with a disjoint variable condition, which requires fewer axioms . (Contributed by NM, 18-Mar-2013.) (Revised by Gino Giotto, 30-Sep-2024.)

Hypothesis

Ref	Expression
cbvriotavw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvriotavw	⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)

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

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

Proof of Theorem cbvriotavw

Step	Hyp	Ref	Expression
1		eleq1w 2821	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2		cbvriotavw.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	anbi12d 631	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓)))
4	3	cbviotavw 6399	. 2 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
5		df-riota 7232	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
6		df-riota 7232	. 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
7	4, 5, 6	3eqtr4i 2776	1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ℩cio 6389  ℩crio 7231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-uni 4840  df-iota 6391  df-riota 7232

This theorem is referenced by:  ordtypecbv  9276  fin23lem27  10084  zorn2g  10259  uspgredg2v  27591  usgredg2v  27594  cnlnadji  30438  nmopadjlei  30450  cvmliftlem15  33260  cvmliftiota  33263  cvmlift2  33278  cvmlift3lem7  33287  cvmlift3  33290  nosupcbv  33905  noinfcbv  33920  lshpkrlem3  37126  cdleme40v  38483  lcfl7N  39515  lcf1o  39565  lcfrlem39  39595  hdmap1cbv  39816  wessf1ornlem  42722  fourierdlem103  43750  fourierdlem104  43751