Theorem cbvriotavw 7398

Description: Change bound variable in a restricted description binder. Version of cbvriotav 7402 with a disjoint variable condition, which requires fewer axioms . (Contributed by NM, 18-Mar-2013.) (Revised by GG, 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 2824	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2		cbvriotavw.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	anbi12d 632	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓)))
4	3	cbviotavw 6522	. 2 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
5		df-riota 7388	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
6		df-riota 7388	. 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
7	4, 5, 6	3eqtr4i 2775	1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ℩cio 6512  ℩crio 7387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-ss 3968  df-uni 4908  df-iota 6514  df-riota 7388

This theorem is referenced by:  ordtypecbv  9557  fin23lem27  10368  zorn2g  10543  nosupcbv  27747  noinfcbv  27762  uspgredg2v  29241  usgredg2v  29244  cnlnadji  32095  nmopadjlei  32107  cvmliftlem15  35303  cvmliftiota  35306  cvmlift2  35321  cvmlift3lem7  35330  cvmlift3  35333  weiunlem2  36464  lshpkrlem3  39113  cdleme40v  40471  lcfl7N  41503  lcf1o  41553  lcfrlem39  41583  hdmap1cbv  41804  wessf1ornlem  45190  fourierdlem103  46224  fourierdlem104  46225