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Theorem ralxfr 4394
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 275 . . 3 ((⊤ ∧ 𝑦𝐶) → 𝐴𝐵)
3 ralxfr.2 . . . 4 (𝑥𝐵 → ∃𝑦𝐶 𝑥 = 𝐴)
43adantl 275 . . 3 ((⊤ ∧ 𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
5 ralxfr.3 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
65adantl 275 . . 3 ((⊤ ∧ 𝑥 = 𝐴) → (𝜑𝜓))
72, 4, 6ralxfrd 4390 . 2 (⊤ → (∀𝑥𝐵 𝜑 ↔ ∀𝑦𝐶 𝜓))
87mptru 1341 1 (∀𝑥𝐵 𝜑 ↔ ∀𝑦𝐶 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1332  wtru 1333  wcel 1481  wral 2417  wrex 2418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691
This theorem is referenced by: (None)
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