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Theorem sbabel 2515
 Description: Theorem to move a substitution in and out of a class abstraction. (Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypothesis
Ref Expression
sbabel.1 xA
Assertion
Ref Expression
sbabel ([y / x]{z φ} A ↔ {z [y / x]φ} A)
Distinct variable groups:   x,z   y,z
Allowed substitution hints:   φ(x,y,z)   A(x,y,z)

Proof of Theorem sbabel
Dummy variable v is distinct from all other variables.
StepHypRef Expression
1 sbex 2128 . . 3 ([y / x]v(v = {z φ} v A) ↔ v[y / x](v = {z φ} v A))
2 sban 2069 . . . . 5 ([y / x](v = {z φ} v A) ↔ ([y / x]v = {z φ} [y / x]v A))
3 nfv 1619 . . . . . . . . . 10 x z v
43sbf 2026 . . . . . . . . 9 ([y / x]z vz v)
54sbrbis 2073 . . . . . . . 8 ([y / x](z vφ) ↔ (z v ↔ [y / x]φ))
65sbalv 2129 . . . . . . 7 ([y / x]z(z vφ) ↔ z(z v ↔ [y / x]φ))
7 abeq2 2458 . . . . . . . 8 (v = {z φ} ↔ z(z vφ))
87sbbii 1653 . . . . . . 7 ([y / x]v = {z φ} ↔ [y / x]z(z vφ))
9 abeq2 2458 . . . . . . 7 (v = {z [y / x]φ} ↔ z(z v ↔ [y / x]φ))
106, 8, 93bitr4i 268 . . . . . 6 ([y / x]v = {z φ} ↔ v = {z [y / x]φ})
11 sbabel.1 . . . . . . . 8 xA
1211nfcri 2483 . . . . . . 7 x v A
1312sbf 2026 . . . . . 6 ([y / x]v Av A)
1410, 13anbi12i 678 . . . . 5 (([y / x]v = {z φ} [y / x]v A) ↔ (v = {z [y / x]φ} v A))
152, 14bitri 240 . . . 4 ([y / x](v = {z φ} v A) ↔ (v = {z [y / x]φ} v A))
1615exbii 1582 . . 3 (v[y / x](v = {z φ} v A) ↔ v(v = {z [y / x]φ} v A))
171, 16bitri 240 . 2 ([y / x]v(v = {z φ} v A) ↔ v(v = {z [y / x]φ} v A))
18 df-clel 2349 . . 3 ({z φ} Av(v = {z φ} v A))
1918sbbii 1653 . 2 ([y / x]{z φ} A ↔ [y / x]v(v = {z φ} v A))
20 df-clel 2349 . 2 ({z [y / x]φ} Av(v = {z [y / x]φ} v A))
2117, 19, 203bitr4i 268 1 ([y / x]{z φ} A ↔ {z [y / x]φ} A)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2476 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478 This theorem is referenced by: (None)
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