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Theorem ralxfrd2 5303
Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of ralxfrd 5299. (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 1110 . . . 4 (((𝜑𝑦𝐶) ∧ 𝑥 = 𝐴) → (𝜓𝜒))
41, 3rspcdv 3612 . . 3 ((𝜑𝑦𝐶) → (∀𝑥𝐵 𝜓𝜒))
54ralrimdva 3186 . 2 (𝜑 → (∀𝑥𝐵 𝜓 → ∀𝑦𝐶 𝜒))
6 ralxfrd2.2 . . . 4 ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
7 r19.29 3251 . . . . 5 ((∀𝑦𝐶 𝜒 ∧ ∃𝑦𝐶 𝑥 = 𝐴) → ∃𝑦𝐶 (𝜒𝑥 = 𝐴))
82ad4ant134 1166 . . . . . . . 8 ((((𝜑𝑥𝐵) ∧ 𝑦𝐶) ∧ 𝑥 = 𝐴) → (𝜓𝜒))
98exbiri 807 . . . . . . 7 (((𝜑𝑥𝐵) ∧ 𝑦𝐶) → (𝑥 = 𝐴 → (𝜒𝜓)))
109impcomd 412 . . . . . 6 (((𝜑𝑥𝐵) ∧ 𝑦𝐶) → ((𝜒𝑥 = 𝐴) → 𝜓))
1110rexlimdva 3281 . . . . 5 ((𝜑𝑥𝐵) → (∃𝑦𝐶 (𝜒𝑥 = 𝐴) → 𝜓))
127, 11syl5 34 . . . 4 ((𝜑𝑥𝐵) → ((∀𝑦𝐶 𝜒 ∧ ∃𝑦𝐶 𝑥 = 𝐴) → 𝜓))
136, 12mpan2d 690 . . 3 ((𝜑𝑥𝐵) → (∀𝑦𝐶 𝜒𝜓))
1413ralrimdva 3186 . 2 (𝜑 → (∀𝑦𝐶 𝜒 → ∀𝑥𝐵 𝜓))
155, 14impbid 213 1 (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1081  df-ex 1772  df-cleq 2811  df-clel 2890  df-ral 3140  df-rex 3141
This theorem is referenced by:  rexxfrd2  5304  ntrclsiso  40295  ntrclsk2  40296  ntrclskb  40297  ntrclsk3  40298  ntrclsk13  40299  ntrclsk4  40300
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