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Theorem cbvrexfw 3304
Description: Rule used to change bound variables, using implicit substitution. Version of cbvrexf 3360 with a disjoint variable condition, which does not require ax-13 2376. For a version not dependent on ax-11 2156 and ax-12, see cbvrexvw 3237. (Contributed by FL, 27-Apr-2008.) Avoid ax-10 2140, ax-13 2376. (Revised by GG, 10-Jan-2024.)
Hypotheses
Ref Expression
cbvrexfw.1 𝑥𝐴
cbvrexfw.2 𝑦𝐴
cbvrexfw.3 𝑦𝜑
cbvrexfw.4 𝑥𝜓
cbvrexfw.5 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvrexfw (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem cbvrexfw
StepHypRef Expression
1 cbvrexfw.1 . . . 4 𝑥𝐴
2 cbvrexfw.2 . . . 4 𝑦𝐴
3 cbvrexfw.3 . . . . 5 𝑦𝜑
43nfn 1856 . . . 4 𝑦 ¬ 𝜑
5 cbvrexfw.4 . . . . 5 𝑥𝜓
65nfn 1856 . . . 4 𝑥 ¬ 𝜓
7 cbvrexfw.5 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
87notbid 318 . . . 4 (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
91, 2, 4, 6, 8cbvralfw 3303 . . 3 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑦𝐴 ¬ 𝜓)
10 ralnex 3071 . . 3 (∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 𝜑)
11 ralnex 3071 . . 3 (∀𝑦𝐴 ¬ 𝜓 ↔ ¬ ∃𝑦𝐴 𝜓)
129, 10, 113bitr3i 301 . 2 (¬ ∃𝑥𝐴 𝜑 ↔ ¬ ∃𝑦𝐴 𝜓)
1312con4bii 321 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wnf 1782  wnfc 2889  wral 3060  wrex 3069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-11 2156  ax-12 2176
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1779  df-nf 1783  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070
This theorem is referenced by:  cbvrexw  3306  reusv2lem4  5400  reusv2  5402  nnwof  12957  cbviunf  32569  ac6sf2  32635  dfimafnf  32647  aciunf1lem  32673  bnj1400  34850  phpreu  37612  poimirlem26  37654  indexa  37741  evth2f  45025  fvelrnbf  45028  evthf  45037  eliin2f  45114  stoweidlem34  46054  ovnlerp  46582
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