Theorem cbvriotavw 7124

Description: Change bound variable in a restricted description binder. Version of cbvriotav 7128 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by NM, 18-Mar-2013.) (Revised by Gino Giotto, 26-Jan-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		nfv 1915	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1915	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvriotavw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvriotaw 7123	1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1537  ℩crio 7113

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-in 3943  df-ss 3952  df-sn 4568  df-uni 4839  df-iota 6314  df-riota 7114

This theorem is referenced by:  ordtypecbv  8981  fin23lem27  9750  zorn2g  9925  uspgredg2v  27006  usgredg2v  27009  cnlnadji  29853  nmopadjlei  29865  cvmliftlem15  32545  cvmliftiota  32548  cvmlift2  32563  cvmlift3lem7  32572  cvmlift3  32575  lshpkrlem3  36263  cdleme40v  37620  lcfl7N  38652  lcf1o  38702  lcfrlem39  38732  hdmap1cbv  38953  wessf1ornlem  41494  fourierdlem103  42543  fourierdlem104  42544