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Theorem nfexd 1734
Description: If 𝑥 is not free in 𝜑, it is not free in 𝑦𝜑. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon, 7-Feb-2018.)
Hypotheses
Ref Expression
nfald.1 𝑦𝜑
nfald.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfexd (𝜑 → Ⅎ𝑥𝑦𝜓)

Proof of Theorem nfexd
StepHypRef Expression
1 nfald.1 . . . . . . 7 𝑦𝜑
21nfri 1499 . . . . . 6 (𝜑 → ∀𝑦𝜑)
3 nfald.2 . . . . . . 7 (𝜑 → Ⅎ𝑥𝜓)
4 df-nf 1437 . . . . . . 7 (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
53, 4sylib 121 . . . . . 6 (𝜑 → ∀𝑥(𝜓 → ∀𝑥𝜓))
62, 5alrimih 1445 . . . . 5 (𝜑 → ∀𝑦𝑥(𝜓 → ∀𝑥𝜓))
7 alcom 1454 . . . . 5 (∀𝑦𝑥(𝜓 → ∀𝑥𝜓) ↔ ∀𝑥𝑦(𝜓 → ∀𝑥𝜓))
86, 7sylib 121 . . . 4 (𝜑 → ∀𝑥𝑦(𝜓 → ∀𝑥𝜓))
9 exim 1578 . . . . 5 (∀𝑦(𝜓 → ∀𝑥𝜓) → (∃𝑦𝜓 → ∃𝑦𝑥𝜓))
109alimi 1431 . . . 4 (∀𝑥𝑦(𝜓 → ∀𝑥𝜓) → ∀𝑥(∃𝑦𝜓 → ∃𝑦𝑥𝜓))
118, 10syl 14 . . 3 (𝜑 → ∀𝑥(∃𝑦𝜓 → ∃𝑦𝑥𝜓))
12 19.12 1643 . . . . 5 (∃𝑦𝑥𝜓 → ∀𝑥𝑦𝜓)
1312imim2i 12 . . . 4 ((∃𝑦𝜓 → ∃𝑦𝑥𝜓) → (∃𝑦𝜓 → ∀𝑥𝑦𝜓))
1413alimi 1431 . . 3 (∀𝑥(∃𝑦𝜓 → ∃𝑦𝑥𝜓) → ∀𝑥(∃𝑦𝜓 → ∀𝑥𝑦𝜓))
1511, 14syl 14 . 2 (𝜑 → ∀𝑥(∃𝑦𝜓 → ∀𝑥𝑦𝜓))
16 df-nf 1437 . 2 (Ⅎ𝑥𝑦𝜓 ↔ ∀𝑥(∃𝑦𝜓 → ∀𝑥𝑦𝜓))
1715, 16sylibr 133 1 (𝜑 → Ⅎ𝑥𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1329  wnf 1436  wex 1468
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-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-nf 1437
This theorem is referenced by:  nfsbxy  1913  nfsbxyt  1914  nfeudv  2012  nfmod  2014  nfeld  2295  nfrexdxy  2466
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