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Theorem cbvrexf 3352
 Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2380. Use the weaker cbvrexfw 3350 when possible. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.)
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 1859 . . . 4 𝑦 ¬ 𝜑
5 cbvralf.4 . . . . 5 𝑥𝜓
65nfn 1859 . . . 4 𝑥 ¬ 𝜓
7 cbvralf.5 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
87notbid 322 . . . 4 (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
91, 2, 4, 6, 8cbvralf 3351 . . 3 (∀𝑥𝐴 ¬ 𝜑 ↔ ∀𝑦𝐴 ¬ 𝜓)
109notbii 324 . 2 (¬ ∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∀𝑦𝐴 ¬ 𝜓)
11 dfrex2 3167 . 2 (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑)
12 dfrex2 3167 . 2 (∃𝑦𝐴 𝜓 ↔ ¬ ∀𝑦𝐴 ¬ 𝜓)
1310, 11, 123bitr4i 307 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  Ⅎwnf 1786  Ⅎwnfc 2900  ∀wral 3071  ∃wrex 3072 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-10 2143  ax-11 2159  ax-12 2176  ax-13 2380 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077 This theorem is referenced by:  cbvrex  3359
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