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Theorem nfexd 1688
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 1455 . . . . . 6 (𝜑 → ∀𝑦𝜑)
3 nfald.2 . . . . . . 7 (𝜑 → Ⅎ𝑥𝜓)
4 df-nf 1393 . . . . . . 7 (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
53, 4sylib 120 . . . . . 6 (𝜑 → ∀𝑥(𝜓 → ∀𝑥𝜓))
62, 5alrimih 1401 . . . . 5 (𝜑 → ∀𝑦𝑥(𝜓 → ∀𝑥𝜓))
7 alcom 1410 . . . . 5 (∀𝑦𝑥(𝜓 → ∀𝑥𝜓) ↔ ∀𝑥𝑦(𝜓 → ∀𝑥𝜓))
86, 7sylib 120 . . . 4 (𝜑 → ∀𝑥𝑦(𝜓 → ∀𝑥𝜓))
9 exim 1533 . . . . 5 (∀𝑦(𝜓 → ∀𝑥𝜓) → (∃𝑦𝜓 → ∃𝑦𝑥𝜓))
109alimi 1387 . . . 4 (∀𝑥𝑦(𝜓 → ∀𝑥𝜓) → ∀𝑥(∃𝑦𝜓 → ∃𝑦𝑥𝜓))
118, 10syl 14 . . 3 (𝜑 → ∀𝑥(∃𝑦𝜓 → ∃𝑦𝑥𝜓))
12 19.12 1598 . . . . 5 (∃𝑦𝑥𝜓 → ∀𝑥𝑦𝜓)
1312imim2i 12 . . . 4 ((∃𝑦𝜓 → ∃𝑦𝑥𝜓) → (∃𝑦𝜓 → ∀𝑥𝑦𝜓))
1413alimi 1387 . . 3 (∀𝑥(∃𝑦𝜓 → ∃𝑦𝑥𝜓) → ∀𝑥(∃𝑦𝜓 → ∀𝑥𝑦𝜓))
1511, 14syl 14 . 2 (𝜑 → ∀𝑥(∃𝑦𝜓 → ∀𝑥𝑦𝜓))
16 df-nf 1393 . 2 (Ⅎ𝑥𝑦𝜓 ↔ ∀𝑥(∃𝑦𝜓 → ∀𝑥𝑦𝜓))
1715, 16sylibr 132 1 (𝜑 → Ⅎ𝑥𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1285  wnf 1392  wex 1424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-nf 1393
This theorem is referenced by:  nfsbxy  1863  nfsbxyt  1864  nfeudv  1960  nfmod  1962  nfeld  2240  nfrexdxy  2407
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