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Theorem rexxfrd 5301
Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by FL, 10-Apr-2007.) (Revised by Mario Carneiro, 15-Aug-2014.)
Hypotheses
Ref Expression
ralxfrd.1 ((𝜑𝑦𝐶) → 𝐴𝐵)
ralxfrd.2 ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
ralxfrd.3 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rexxfrd (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵   𝑥,𝐶   𝜒,𝑥   𝜑,𝑥,𝑦   𝜓,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑦)   𝐶(𝑦)

Proof of Theorem rexxfrd
StepHypRef Expression
1 ralxfrd.1 . . . 4 ((𝜑𝑦𝐶) → 𝐴𝐵)
2 ralxfrd.2 . . . 4 ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
3 ralxfrd.3 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
43notbid 320 . . . 4 ((𝜑𝑥 = 𝐴) → (¬ 𝜓 ↔ ¬ 𝜒))
51, 2, 4ralxfrd 5300 . . 3 (𝜑 → (∀𝑥𝐵 ¬ 𝜓 ↔ ∀𝑦𝐶 ¬ 𝜒))
65notbid 320 . 2 (𝜑 → (¬ ∀𝑥𝐵 ¬ 𝜓 ↔ ¬ ∀𝑦𝐶 ¬ 𝜒))
7 dfrex2 3239 . 2 (∃𝑥𝐵 𝜓 ↔ ¬ ∀𝑥𝐵 ¬ 𝜓)
8 dfrex2 3239 . 2 (∃𝑦𝐶 𝜒 ↔ ¬ ∀𝑦𝐶 ¬ 𝜒)
96, 7, 83bitr4g 316 1 (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  wral 3138  wrex 3139
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-cleq 2814  df-clel 2893  df-ral 3143  df-rex 3144
This theorem is referenced by:  cmpfi  22010  elfm  22549  rlimcnp  25537  rmoxfrd  30251  iunrdx  30309  swrdrn3  30624  dvh4dimat  38568  mapdcv  38790  elrfirn  39285  fargshiftfo  43596
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