Theorem cbvrex2vw 3240

Description: Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvrex2v 3367 with a disjoint variable condition, which does not require ax-13 2375. (Contributed by FL, 2-Jul-2012.) Avoid ax-13 2375. (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 3177	. . 3 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜒))
3	2	cbvrexvw 3236	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)
4		cbvrex2vw.2	. . . 4 ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))
5	4	cbvrexvw 3236	. . 3 ⊢ (∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑤 ∈ 𝐵 𝜓)
6	5	rexbii 3092	. 2 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓)
7	3, 6	bitri 275	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 206  ∃wrex 3068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-clel 2814  df-rex 3069

This theorem is referenced by:  omeu  8622  oeeui  8639  eroveu  8851  genpv  11037  bezoutlem3  16575  bezoutlem4  16576  bezout  16577  4sqlem2  16983  vdwnn  17032  efgrelexlema  19782  dyadmax  25647  2sqlem9  27486  2sq  27489  mulsval2lem  28151  mulsunif2  28211  precsexlemcbv  28245  legov  28608  dfcgra2  28853  gsumwun  33051  pstmfval  33857  satfv0  35343  satfv0fun  35356  fmla1  35372  nn0prpwlem  36305  isbnd2  37770  hashnexinjle  42111  aks6d1c6lem3  42154  nna4b4nsq  42647  oaun3lem1  43364  limsupref  45641  fourierdlem42  46105  fourierdlem54  46116  mogoldbb  47710