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Theorem ralxfrd 5414
Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) (Proof shortened by JJ, 7-Aug-2021.)
Hypotheses
Ref Expression
ralxfrd.1 ((𝜑𝑦𝐶) → 𝐴𝐵)
ralxfrd.2 ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
ralxfrd.3 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
ralxfrd (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 ralxfrd
StepHypRef Expression
1 ralxfrd.1 . . . 4 ((𝜑𝑦𝐶) → 𝐴𝐵)
2 ralxfrd.3 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
32adantlr 715 . . . 4 (((𝜑𝑦𝐶) ∧ 𝑥 = 𝐴) → (𝜓𝜒))
41, 3rspcdv 3614 . . 3 ((𝜑𝑦𝐶) → (∀𝑥𝐵 𝜓𝜒))
54ralrimdva 3152 . 2 (𝜑 → (∀𝑥𝐵 𝜓 → ∀𝑦𝐶 𝜒))
6 ralxfrd.2 . . . 4 ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
7 r19.29 3112 . . . . . 6 ((∀𝑦𝐶 𝜒 ∧ ∃𝑦𝐶 𝑥 = 𝐴) → ∃𝑦𝐶 (𝜒𝑥 = 𝐴))
82exbiri 811 . . . . . . . 8 (𝜑 → (𝑥 = 𝐴 → (𝜒𝜓)))
98impcomd 411 . . . . . . 7 (𝜑 → ((𝜒𝑥 = 𝐴) → 𝜓))
109rexlimdvw 3158 . . . . . 6 (𝜑 → (∃𝑦𝐶 (𝜒𝑥 = 𝐴) → 𝜓))
117, 10syl5 34 . . . . 5 (𝜑 → ((∀𝑦𝐶 𝜒 ∧ ∃𝑦𝐶 𝑥 = 𝐴) → 𝜓))
1211adantr 480 . . . 4 ((𝜑𝑥𝐵) → ((∀𝑦𝐶 𝜒 ∧ ∃𝑦𝐶 𝑥 = 𝐴) → 𝜓))
136, 12mpan2d 694 . . 3 ((𝜑𝑥𝐵) → (∀𝑦𝐶 𝜒𝜓))
1413ralrimdva 3152 . 2 (𝜑 → (∀𝑦𝐶 𝜒 → ∀𝑥𝐵 𝜓))
155, 14impbid 212 1 (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  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-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069
This theorem is referenced by:  rexxfrd  5415  ralxfr2d  5416  ralxfr  5420  islindf4  21876  cmpfi  23432  rlimcnp  27023  ispisys2  34134  glbconN  39359  glbconNOLD  39360  mapdordlem2  41620  opncldeqv  48698
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