Theorem cbvrex2vw 3249

Description: Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvrex2v 3358 with a disjoint variable condition, which does not require ax-13 2404. (Contributed by FL, 2-Jul-2012.) Avoid ax-13 2404. (Revised by GG, 10-Jan-2024.)

Hypotheses

Ref	Expression
cbvrex2vw.1	⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒))
cbvrex2vw.2	⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))

Assertion

Ref	Expression
cbvrex2vw	⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓)

Distinct variable groups:   𝑥,𝑧   𝑦,𝑤   𝑥,𝐴,𝑧   𝑤,𝐵   𝑥,𝐵,𝑦,𝑧   𝜒,𝑤   𝜒,𝑥   𝜑,𝑧   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑤)   𝜓(𝑥, 𝑧, 𝑤)   𝜒(𝑦, 𝑧)   𝐴(𝑦, 𝑤)

Proof of Theorem cbvrex2vw

Step	Hyp	Ref	Expression
1		cbvrex2vw.1	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒))
2	1	rexbidv 3190	. . 3 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜒))
3	2	cbvrexvw 3245	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)
4		cbvrex2vw.2	. . . 4 ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))
5	4	cbvrexvw 3245	. . 3 ⊢ (∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑤 ∈ 𝐵 𝜓)
6	5	rexbii 3113	. 2 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓)
7	3, 6	bitri 278	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∃wrex 3090

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-clel 2839  df-rex 3091

This theorem is used by:  omeu  8631  oeeui  8649  eroveu  8871  genpv  11123  bezoutlem3  16758  bezoutlem4  16759  bezout  16760  4sqlem2  17171  vdwnn  17220  efgrelexlema  20014  dyadmax  25974  2sqlem9  27811  2sq  27814  nna4b4nsq  28047  mulsval2lem  28553  mulsunif2  28613  precsexlemcbv  28649  eucliddivs  28819  bdayfinbndcbv  28909  bdayfinbndlem1  28910  bdayfinbndlem2  28911  z12zsodd  28925  legov  29105  dfcgra2  29395  gsumwun  33736  constrcbvlem  34498  pstmfval  34639  satfv0  36331  satfv0fun  36344  fmla1  36360  nn0prpwlem  37325  isbnd2  38983  hashnexinjle  43445  aks6d1c6lem3  43488  oaun3lem1  44637  limsupref  46968  fourierdlem42  47432  fourierdlem54  47443  mogoldbb  49156