Theorem cbvrex2v 3156

Description: Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.)

Hypotheses

Ref	Expression
cbvrex2v.1	⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒))
cbvrex2v.2	⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))

Assertion

Ref	Expression
cbvrex2v	⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓)

Distinct variable groups:   𝑥,𝐴   𝑧,𝐴   𝑤,𝐵   𝑥,𝐵,𝑦   𝑧,𝐵,𝑦   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 cbvrex2v

Step	Hyp	Ref	Expression
1		cbvrex2v.1	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒))
2	1	rexbidv 3034	. . 3 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜒))
3	2	cbvrexv 3148	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)
4		cbvrex2v.2	. . . 4 ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))
5	4	cbvrexv 3148	. . 3 ⊢ (∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑤 ∈ 𝐵 𝜓)
6	5	rexbii 3023	. 2 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓)
7	3, 6	bitri 263	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 195  ∃wrex 2897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902

This theorem is referenced by:  omeu  7552  oeeui  7569  eroveu  7729  genpv  9700  bezoutlem3  15096  bezoutlem4  15097  bezout  15098  4sqlem2  15491  vdwnn  15540  efgrelexlema  17985  dyadmax  23172  2sqlem9  24952  2sq  24955  legov  25280  dfcgra2  25521  pstmfval  29267  nn0prpwlem  31487  isbnd2  32752  fourierdlem42  39042  fourierdlem54  39053