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Theorem cbvrexfw 3287
Description: Rule used to change bound variables, using implicit substitution. Version of cbvrexf 3333 with a disjoint variable condition, which does not require ax-13 2371. For a version not dependent on ax-11 2155 and ax-12, see cbvrexvw 3225. (Contributed by FL, 27-Apr-2008.) Avoid ax-10 2138, ax-13 2371. (Revised by Gino Giotto, 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 1861 . . . 4 𝑦 ¬ 𝜑
5 cbvrexfw.4 . . . . 5 𝑥𝜓
65nfn 1861 . . . 4 𝑥 ¬ 𝜓
7 cbvrexfw.5 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
87notbid 318 . . . 4 (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
91, 2, 4, 6, 8cbvralfw 3286 . . 3 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑦𝐴 ¬ 𝜓)
10 ralnex 3072 . . 3 (∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 𝜑)
11 ralnex 3072 . . 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 205  wnf 1786  wnfc 2884  wral 3061  wrex 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-11 2155  ax-12 2172
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071
This theorem is referenced by:  cbvrexw  3289  reusv2lem4  5357  reusv2  5359  nnwof  12844  cbviunf  31520  ac6sf2  31585  dfimafnf  31596  aciunf1lem  31624  bnj1400  33504  phpreu  36108  poimirlem26  36150  indexa  36238  evth2f  43308  fvelrnbf  43311  evthf  43320  eliin2f  43402  stoweidlem34  44361  ovnlerp  44889
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