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Theorem ralxfrd 4909
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 751 . . . 4 (((𝜑𝑦𝐶) ∧ 𝑥 = 𝐴) → (𝜓𝜒))
41, 3rspcdv 3343 . . 3 ((𝜑𝑦𝐶) → (∀𝑥𝐵 𝜓𝜒))
54ralrimdva 2998 . 2 (𝜑 → (∀𝑥𝐵 𝜓 → ∀𝑦𝐶 𝜒))
6 ralxfrd.2 . . . 4 ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
7 r19.29 3101 . . . . . 6 ((∀𝑦𝐶 𝜒 ∧ ∃𝑦𝐶 𝑥 = 𝐴) → ∃𝑦𝐶 (𝜒𝑥 = 𝐴))
82exbiri 651 . . . . . . . . 9 (𝜑 → (𝑥 = 𝐴 → (𝜒𝜓)))
98com23 86 . . . . . . . 8 (𝜑 → (𝜒 → (𝑥 = 𝐴𝜓)))
109impd 446 . . . . . . 7 (𝜑 → ((𝜒𝑥 = 𝐴) → 𝜓))
1110rexlimdvw 3063 . . . . . 6 (𝜑 → (∃𝑦𝐶 (𝜒𝑥 = 𝐴) → 𝜓))
127, 11syl5 34 . . . . 5 (𝜑 → ((∀𝑦𝐶 𝜒 ∧ ∃𝑦𝐶 𝑥 = 𝐴) → 𝜓))
1312adantr 480 . . . 4 ((𝜑𝑥𝐵) → ((∀𝑦𝐶 𝜒 ∧ ∃𝑦𝐶 𝑥 = 𝐴) → 𝜓))
146, 13mpan2d 710 . . 3 ((𝜑𝑥𝐵) → (∀𝑦𝐶 𝜒𝜓))
1514ralrimdva 2998 . 2 (𝜑 → (∀𝑦𝐶 𝜒 → ∀𝑥𝐵 𝜓))
165, 15impbid 202 1 (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233
This theorem is referenced by:  rexxfrd  4911  ralxfr2d  4912  ralxfr  4916  islindf4  20225  cmpfi  21259  rlimcnp  24737  ispisys2  30344  glbconN  34981  mapdordlem2  37243
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