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Theorem ralxfr2d 5080
Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Mario Carneiro, 20-Aug-2014.)
Hypotheses
Ref Expression
ralxfr2d.1 ((𝜑𝑦𝐶) → 𝐴𝑉)
ralxfr2d.2 (𝜑 → (𝑥𝐵 ↔ ∃𝑦𝐶 𝑥 = 𝐴))
ralxfr2d.3 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
ralxfr2d (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)   𝑉(𝑥,𝑦)

Proof of Theorem ralxfr2d
StepHypRef Expression
1 ralxfr2d.1 . . . 4 ((𝜑𝑦𝐶) → 𝐴𝑉)
2 elisset 3403 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
31, 2syl 17 . . 3 ((𝜑𝑦𝐶) → ∃𝑥 𝑥 = 𝐴)
4 ralxfr2d.2 . . . . . . . 8 (𝜑 → (𝑥𝐵 ↔ ∃𝑦𝐶 𝑥 = 𝐴))
54biimprd 240 . . . . . . 7 (𝜑 → (∃𝑦𝐶 𝑥 = 𝐴𝑥𝐵))
6 r19.23v 3204 . . . . . . 7 (∀𝑦𝐶 (𝑥 = 𝐴𝑥𝐵) ↔ (∃𝑦𝐶 𝑥 = 𝐴𝑥𝐵))
75, 6sylibr 226 . . . . . 6 (𝜑 → ∀𝑦𝐶 (𝑥 = 𝐴𝑥𝐵))
87r19.21bi 3113 . . . . 5 ((𝜑𝑦𝐶) → (𝑥 = 𝐴𝑥𝐵))
9 eleq1 2866 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
108, 9mpbidi 233 . . . 4 ((𝜑𝑦𝐶) → (𝑥 = 𝐴𝐴𝐵))
1110exlimdv 2029 . . 3 ((𝜑𝑦𝐶) → (∃𝑥 𝑥 = 𝐴𝐴𝐵))
123, 11mpd 15 . 2 ((𝜑𝑦𝐶) → 𝐴𝐵)
134biimpa 469 . 2 ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
14 ralxfr2d.3 . 2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
1512, 13, 14ralxfrd 5078 1 (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wex 1875  wcel 2157  wral 3089  wrex 3090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-v 3387
This theorem is referenced by:  rexxfr2d  5081  ralrn  6588  ralima  6727  cnrest2  21419  cnprest2  21423  connsuba  21552  subislly  21613  trfbas2  21975  trfil2  22019  flimrest  22115  fclsrest  22156  tsmssubm  22274  metucn  22704  extoimad  39246
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