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Theorem nfsb4t 2080
 Description: A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 2081). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.)
Assertion
Ref Expression
nfsb4t (xzφ → (¬ z z = y → Ⅎz[y / x]φ))

Proof of Theorem nfsb4t
StepHypRef Expression
1 sbequ12 1919 . . . . . . . . 9 (x = y → (φ ↔ [y / x]φ))
21sps 1754 . . . . . . . 8 (x x = y → (φ ↔ [y / x]φ))
32drnf2 1970 . . . . . . 7 (x x = y → (Ⅎzφ ↔ Ⅎz[y / x]φ))
43biimpcd 215 . . . . . 6 (Ⅎzφ → (x x = y → Ⅎz[y / x]φ))
54sps 1754 . . . . 5 (xzφ → (x x = y → Ⅎz[y / x]φ))
65a1dd 42 . . . 4 (xzφ → (x x = y → ((¬ z z = x ¬ z z = y) → Ⅎz[y / x]φ)))
7 nfa1 1788 . . . . . . . 8 xxzφ
8 nfnae 1956 . . . . . . . . 9 x ¬ z z = x
9 nfnae 1956 . . . . . . . . 9 x ¬ z z = y
108, 9nfan 1824 . . . . . . . 8 xz z = x ¬ z z = y)
117, 10nfan 1824 . . . . . . 7 x(xzφ z z = x ¬ z z = y))
12 nfeqf 1958 . . . . . . . . 9 ((¬ z z = x ¬ z z = y) → Ⅎz x = y)
1312adantl 452 . . . . . . . 8 ((xzφ z z = x ¬ z z = y)) → Ⅎz x = y)
14 sp 1747 . . . . . . . . 9 (xzφ → Ⅎzφ)
1514adantr 451 . . . . . . . 8 ((xzφ z z = x ¬ z z = y)) → Ⅎzφ)
1613, 15nfimd 1808 . . . . . . 7 ((xzφ z z = x ¬ z z = y)) → Ⅎz(x = yφ))
1711, 16nfald 1852 . . . . . 6 ((xzφ z z = x ¬ z z = y)) → Ⅎzx(x = yφ))
1817ex 423 . . . . 5 (xzφ → ((¬ z z = x ¬ z z = y) → Ⅎzx(x = yφ)))
19 nfnae 1956 . . . . . . 7 z ¬ x x = y
20 sb4b 2054 . . . . . . 7 x x = y → ([y / x]φx(x = yφ)))
2119, 20nfbidf 1774 . . . . . 6 x x = y → (Ⅎz[y / x]φ ↔ Ⅎzx(x = yφ)))
2221imbi2d 307 . . . . 5 x x = y → (((¬ z z = x ¬ z z = y) → Ⅎz[y / x]φ) ↔ ((¬ z z = x ¬ z z = y) → Ⅎzx(x = yφ))))
2318, 22syl5ibrcom 213 . . . 4 (xzφ → (¬ x x = y → ((¬ z z = x ¬ z z = y) → Ⅎz[y / x]φ)))
246, 23pm2.61d 150 . . 3 (xzφ → ((¬ z z = x ¬ z z = y) → Ⅎz[y / x]φ))
2524exp3a 425 . 2 (xzφ → (¬ z z = x → (¬ z z = y → Ⅎz[y / x]φ)))
26 nfsb2 2058 . . 3 z z = y → Ⅎz[y / z]φ)
27 drsb1 2022 . . . 4 (z z = x → ([y / z]φ ↔ [y / x]φ))
2827drnf2 1970 . . 3 (z z = x → (Ⅎz[y / z]φ ↔ Ⅎz[y / x]φ))
2926, 28syl5ib 210 . 2 (z z = x → (¬ z z = y → Ⅎz[y / x]φ))
3025, 29pm2.61d2 152 1 (xzφ → (¬ z z = y → Ⅎz[y / x]φ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544  [wsb 1648 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  nfsb4  2081  dvelimdf  2082  nfsbd  2111
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