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Theorem cbvrexf 3400
 Description: Rule used to change bound variables, using implicit substitution. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.)
Hypotheses
Ref Expression
cbvralf.1 𝑥𝐴
cbvralf.2 𝑦𝐴
cbvralf.3 𝑦𝜑
cbvralf.4 𝑥𝜓
cbvralf.5 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvrexf (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)

Proof of Theorem cbvrexf
StepHypRef Expression
1 cbvralf.1 . . . 4 𝑥𝐴
2 cbvralf.2 . . . 4 𝑦𝐴
3 cbvralf.3 . . . . 5 𝑦𝜑
43nfn 1842 . . . 4 𝑦 ¬ 𝜑
5 cbvralf.4 . . . . 5 𝑥𝜓
65nfn 1842 . . . 4 𝑥 ¬ 𝜓
7 cbvralf.5 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
87notbid 319 . . . 4 (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
91, 2, 4, 6, 8cbvralf 3399 . . 3 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑦𝐴 ¬ 𝜓)
109notbii 321 . 2 (¬ ∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∀𝑦𝐴 ¬ 𝜓)
11 dfrex2 3205 . 2 (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑)
12 dfrex2 3205 . 2 (∃𝑦𝐴 𝜓 ↔ ¬ ∀𝑦𝐴 ¬ 𝜓)
1310, 11, 123bitr4i 304 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207  Ⅎwnf 1769  Ⅎwnfc 2935  ∀wral 3107  ∃wrex 3108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113 This theorem is referenced by:  cbvrex  3402  reusv2lem4  5200  reusv2  5202  nnwof  12167  cbviunf  29993  ac6sf2  30056  dfimafnf  30067  aciunf1lem  30093  bnj1400  31720  phpreu  34428  poimirlem26  34470  indexa  34561  evth2f  40832  fvelrnbf  40835  evthf  40844  eliin2f  40931  stoweidlem34  41883  ovnlerp  42408
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