Theorem cbvrex2vw 3248

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 3189	. . 3 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜒))
3	2	cbvrexvw 3244	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)
4		cbvrex2vw.2	. . . 4 ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))
5	4	cbvrexvw 3244	. . 3 ⊢ (∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑤 ∈ 𝐵 𝜓)
6	5	rexbii 3112	. 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)

Syntax hints:   → wi 4   ↔ wb 209  ∃wrex 3089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-clel 2838  df-rex 3090

This theorem is referenced by:  omeu  8566  oeeui  8584  eroveu  8806  genpv  10979  bezoutlem3  16594  bezoutlem4  16595  bezout  16596  4sqlem2  17004  vdwnn  17053  efgrelexlema  19814  dyadmax  25757  2sqlem9  27591  2sq  27594  mulsval2lem  28303  mulsunif2  28363  precsexlemcbv  28399  eucliddivs  28569  bdayfinbndcbv  28659  bdayfinbndlem1  28660  bdayfinbndlem2  28661  z12zsodd  28675  legov  28854  dfcgra2  29141  gsumwun  33396  constrcbvlem  34145  pstmfval  34286  satfv0  35850  satfv0fun  35863  fmla1  35879  nn0prpwlem  36833  isbnd2  38434  hashnexinjle  42896  aks6d1c6lem3  42939  nna4b4nsq  43392  oaun3lem1  44101  limsupref  46399  fourierdlem42  46863  fourierdlem54  46874  mogoldbb  48550