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Theorem cbvrexfw 3303
Description: Rule used to change bound variables, using implicit substitution. Version of cbvrexf 3359 with a disjoint variable condition, which does not require ax-13 2375. For a version not dependent on ax-11 2155 and ax-12, see cbvrexvw 3236. (Contributed by FL, 27-Apr-2008.) Avoid ax-10 2139, ax-13 2375. (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 1855 . . . 4 𝑦 ¬ 𝜑
5 cbvrexfw.4 . . . . 5 𝑥𝜓
65nfn 1855 . . . 4 𝑥 ¬ 𝜓
7 cbvrexfw.5 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
87notbid 318 . . . 4 (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
91, 2, 4, 6, 8cbvralfw 3302 . . 3 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑦𝐴 ¬ 𝜓)
10 ralnex 3070 . . 3 (∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 𝜑)
11 ralnex 3070 . . 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 1780  wnfc 2888  wral 3059  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  ax-11 2155  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069
This theorem is referenced by:  cbvrexw  3305  reusv2lem4  5407  reusv2  5409  nnwof  12954  cbviunf  32576  ac6sf2  32642  dfimafnf  32653  aciunf1lem  32679  bnj1400  34828  phpreu  37591  poimirlem26  37633  indexa  37720  evth2f  44953  fvelrnbf  44956  evthf  44965  eliin2f  45044  stoweidlem34  45990  ovnlerp  46518
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