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Theorem alxfr 5006
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 3442 . . . . . 6 (𝐴𝐵 → (∀𝑥𝜑𝜓))
32com12 32 . . . . 5 (∀𝑥𝜑 → (𝐴𝐵𝜓))
43alimdv 1996 . . . 4 (∀𝑥𝜑 → (∀𝑦 𝐴𝐵 → ∀𝑦𝜓))
54com12 32 . . 3 (∀𝑦 𝐴𝐵 → (∀𝑥𝜑 → ∀𝑦𝜓))
65adantr 466 . 2 ((∀𝑦 𝐴𝐵 ∧ ∀𝑥𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 → ∀𝑦𝜓))
7 nfa1 2183 . . . . . 6 𝑦𝑦𝜓
8 nfv 1994 . . . . . 6 𝑦𝜑
9 sp 2206 . . . . . . 7 (∀𝑦𝜓𝜓)
109, 1syl5ibrcom 237 . . . . . 6 (∀𝑦𝜓 → (𝑥 = 𝐴𝜑))
117, 8, 10exlimd 2242 . . . . 5 (∀𝑦𝜓 → (∃𝑦 𝑥 = 𝐴𝜑))
1211alimdv 1996 . . . 4 (∀𝑦𝜓 → (∀𝑥𝑦 𝑥 = 𝐴 → ∀𝑥𝜑))
1312com12 32 . . 3 (∀𝑥𝑦 𝑥 = 𝐴 → (∀𝑦𝜓 → ∀𝑥𝜑))
1413adantl 467 . 2 ((∀𝑦 𝐴𝐵 ∧ ∀𝑥𝑦 𝑥 = 𝐴) → (∀𝑦𝜓 → ∀𝑥𝜑))
156, 14impbid 202 1 ((∀𝑦 𝐴𝐵 ∧ ∀𝑥𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1628   = wceq 1630  wex 1851  wcel 2144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351
This theorem is referenced by: (None)
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