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Theorem ralxfr 4812
 Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)
Hypotheses
Ref Expression
ralxfr.1 (𝑦𝐶𝐴𝐵)
ralxfr.2 (𝑥𝐵 → ∃𝑦𝐶 𝑥 = 𝐴)
ralxfr.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ralxfr (∀𝑥𝐵 𝜑 ↔ ∀𝑦𝐶 𝜓)
Distinct variable groups:   𝜓,𝑥   𝜑,𝑦   𝑥,𝐴   𝑥,𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑦)   𝐶(𝑦)

Proof of Theorem ralxfr
StepHypRef Expression
1 ralxfr.1 . . . 4 (𝑦𝐶𝐴𝐵)
21adantl 481 . . 3 ((⊤ ∧ 𝑦𝐶) → 𝐴𝐵)
3 ralxfr.2 . . . 4 (𝑥𝐵 → ∃𝑦𝐶 𝑥 = 𝐴)
43adantl 481 . . 3 ((⊤ ∧ 𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
5 ralxfr.3 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
65adantl 481 . . 3 ((⊤ ∧ 𝑥 = 𝐴) → (𝜑𝜓))
72, 4, 6ralxfrd 4805 . 2 (⊤ → (∀𝑥𝐵 𝜑 ↔ ∀𝑦𝐶 𝜓))
87trud 1484 1 (∀𝑥𝐵 𝜑 ↔ ∀𝑦𝐶 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  ⊤wtru 1476   ∈ 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-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:  rexxfr  4814  infm3  10861
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