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Theorem ralxfrd2 4810
 Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of ralxfrd 4805. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
Hypotheses
Ref Expression
ralxfrd2.1 ((𝜑𝑦𝐶) → 𝐴𝐵)
ralxfrd2.2 ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
ralxfrd2.3 ((𝜑𝑦𝐶𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
ralxfrd2 (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 ralxfrd2
StepHypRef Expression
1 ralxfrd2.1 . . . 4 ((𝜑𝑦𝐶) → 𝐴𝐵)
2 ralxfrd2.3 . . . . 5 ((𝜑𝑦𝐶𝑥 = 𝐴) → (𝜓𝜒))
323expa 1257 . . . 4 (((𝜑𝑦𝐶) ∧ 𝑥 = 𝐴) → (𝜓𝜒))
41, 3rspcdv 3285 . . 3 ((𝜑𝑦𝐶) → (∀𝑥𝐵 𝜓𝜒))
54ralrimdva 2952 . 2 (𝜑 → (∀𝑥𝐵 𝜓 → ∀𝑦𝐶 𝜒))
6 ralxfrd2.2 . . . 4 ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
7 r19.29 3054 . . . . 5 ((∀𝑦𝐶 𝜒 ∧ ∃𝑦𝐶 𝑥 = 𝐴) → ∃𝑦𝐶 (𝜒𝑥 = 𝐴))
82ad4ant134 1288 . . . . . . . . 9 ((((𝜑𝑥𝐵) ∧ 𝑦𝐶) ∧ 𝑥 = 𝐴) → (𝜓𝜒))
98exbiri 650 . . . . . . . 8 (((𝜑𝑥𝐵) ∧ 𝑦𝐶) → (𝑥 = 𝐴 → (𝜒𝜓)))
109com23 84 . . . . . . 7 (((𝜑𝑥𝐵) ∧ 𝑦𝐶) → (𝜒 → (𝑥 = 𝐴𝜓)))
1110impd 446 . . . . . 6 (((𝜑𝑥𝐵) ∧ 𝑦𝐶) → ((𝜒𝑥 = 𝐴) → 𝜓))
1211rexlimdva 3013 . . . . 5 ((𝜑𝑥𝐵) → (∃𝑦𝐶 (𝜒𝑥 = 𝐴) → 𝜓))
137, 12syl5 33 . . . 4 ((𝜑𝑥𝐵) → ((∀𝑦𝐶 𝜒 ∧ ∃𝑦𝐶 𝑥 = 𝐴) → 𝜓))
146, 13mpan2d 706 . . 3 ((𝜑𝑥𝐵) → (∀𝑦𝐶 𝜒𝜓))
1514ralrimdva 2952 . 2 (𝜑 → (∀𝑦𝐶 𝜒 → ∀𝑥𝐵 𝜓))
165, 15impbid 201 1 (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶 𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175 This theorem is referenced by:  rexxfrd2  4811  ntrclsiso  37385  ntrclsk2  37386  ntrclskb  37387  ntrclsk3  37388  ntrclsk13  37389  ntrclsk4  37390
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