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Theorem ralxfrdOLD 4845
Description: Obsolete proof of ralxfrd 4844 as of 7-Aug-2021. (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
ralxfrd.1 ((𝜑𝑦𝐶) → 𝐴𝐵)
ralxfrd.2 ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
ralxfrd.3 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
ralxfrdOLD (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 ralxfrdOLD
StepHypRef Expression
1 ralxfrd.1 . . . 4 ((𝜑𝑦𝐶) → 𝐴𝐵)
2 ralxfrd.3 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
32adantlr 750 . . . 4 (((𝜑𝑦𝐶) ∧ 𝑥 = 𝐴) → (𝜓𝜒))
41, 3rspcdv 3301 . . 3 ((𝜑𝑦𝐶) → (∀𝑥𝐵 𝜓𝜒))
54ralrimdva 2964 . 2 (𝜑 → (∀𝑥𝐵 𝜓 → ∀𝑦𝐶 𝜒))
6 ralxfrd.2 . . . 4 ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
7 r19.29 3066 . . . . 5 ((∀𝑦𝐶 𝜒 ∧ ∃𝑦𝐶 𝑥 = 𝐴) → ∃𝑦𝐶 (𝜒𝑥 = 𝐴))
82biimprd 238 . . . . . . . . 9 ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
98expimpd 628 . . . . . . . 8 (𝜑 → ((𝑥 = 𝐴𝜒) → 𝜓))
109ancomsd 470 . . . . . . 7 (𝜑 → ((𝜒𝑥 = 𝐴) → 𝜓))
1110ad2antrr 761 . . . . . 6 (((𝜑𝑥𝐵) ∧ 𝑦𝐶) → ((𝜒𝑥 = 𝐴) → 𝜓))
1211rexlimdva 3025 . . . . 5 ((𝜑𝑥𝐵) → (∃𝑦𝐶 (𝜒𝑥 = 𝐴) → 𝜓))
137, 12syl5 34 . . . 4 ((𝜑𝑥𝐵) → ((∀𝑦𝐶 𝜒 ∧ ∃𝑦𝐶 𝑥 = 𝐴) → 𝜓))
146, 13mpan2d 709 . . 3 ((𝜑𝑥𝐵) → (∀𝑦𝐶 𝜒𝜓))
1514ralrimdva 2964 . 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 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3191
This theorem is referenced by: (None)
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