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Theorem alxfr 5279
 Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 18-Feb-2007.)
Hypothesis
Ref Expression
alxfr.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
alxfr ((∀𝑦 𝐴𝐵 ∧ ∀𝑥𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem alxfr
StepHypRef Expression
1 alxfr.1 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜓))
21spcgv 3515 . . . . . 6 (𝐴𝐵 → (∀𝑥𝜑𝜓))
32com12 32 . . . . 5 (∀𝑥𝜑 → (𝐴𝐵𝜓))
43alimdv 1917 . . . 4 (∀𝑥𝜑 → (∀𝑦 𝐴𝐵 → ∀𝑦𝜓))
54com12 32 . . 3 (∀𝑦 𝐴𝐵 → (∀𝑥𝜑 → ∀𝑦𝜓))
65adantr 484 . 2 ((∀𝑦 𝐴𝐵 ∧ ∀𝑥𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 → ∀𝑦𝜓))
7 nfa1 2152 . . . . . 6 𝑦𝑦𝜓
8 nfv 1915 . . . . . 6 𝑦𝜑
9 sp 2180 . . . . . . 7 (∀𝑦𝜓𝜓)
109, 1syl5ibrcom 250 . . . . . 6 (∀𝑦𝜓 → (𝑥 = 𝐴𝜑))
117, 8, 10exlimd 2216 . . . . 5 (∀𝑦𝜓 → (∃𝑦 𝑥 = 𝐴𝜑))
1211alimdv 1917 . . . 4 (∀𝑦𝜓 → (∀𝑥𝑦 𝑥 = 𝐴 → ∀𝑥𝜑))
1312com12 32 . . 3 (∀𝑥𝑦 𝑥 = 𝐴 → (∀𝑦𝜓 → ∀𝑥𝜑))
1413adantl 485 . 2 ((∀𝑦 𝐴𝐵 ∧ ∀𝑥𝑦 𝑥 = 𝐴) → (∀𝑦𝜓 → ∀𝑥𝜑))
156, 14impbid 215 1 ((∀𝑦 𝐴𝐵 ∧ ∀𝑥𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411 This theorem is referenced by: (None)
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