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Theorem ralxfr 5209
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 482 . . 3 ((⊤ ∧ 𝑦𝐶) → 𝐴𝐵)
3 ralxfr.2 . . . 4 (𝑥𝐵 → ∃𝑦𝐶 𝑥 = 𝐴)
43adantl 482 . . 3 ((⊤ ∧ 𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
5 ralxfr.3 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
65adantl 482 . . 3 ((⊤ ∧ 𝑥 = 𝐴) → (𝜑𝜓))
72, 4, 6ralxfrd 5203 . 2 (⊤ → (∀𝑥𝐵 𝜑 ↔ ∀𝑦𝐶 𝜓))
87mptru 1529 1 (∀𝑥𝐵 𝜑 ↔ ∀𝑦𝐶 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1522  wtru 1523  wcel 2080  wral 3104  wrex 3105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-ext 2768
This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1525  df-ex 1763  df-sb 2042  df-clab 2775  df-cleq 2787  df-clel 2862  df-ral 3109  df-rex 3110  df-v 3438
This theorem is referenced by:  rexxfr  5211  infm3  11450
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