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Theorem 2sb6rf 2118
 Description: Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypotheses
Ref Expression
2sb5rf.1 zφ
2sb5rf.2 wφ
Assertion
Ref Expression
2sb6rf (φzw((z = x w = y) → [z / x][w / y]φ))
Distinct variable groups:   x,y   x,w   y,z   z,w
Allowed substitution hints:   φ(x,y,z,w)

Proof of Theorem 2sb6rf
StepHypRef Expression
1 2sb5rf.1 . . 3 zφ
21sb6rf 2091 . 2 (φz(z = x → [z / x]φ))
3 19.21v 1890 . . . 4 (w(z = x → (w = y → [w / y][z / x]φ)) ↔ (z = xw(w = y → [w / y][z / x]φ)))
4 sbcom2 2114 . . . . . . 7 ([z / x][w / y]φ ↔ [w / y][z / x]φ)
54imbi2i 303 . . . . . 6 (((z = x w = y) → [z / x][w / y]φ) ↔ ((z = x w = y) → [w / y][z / x]φ))
6 impexp 433 . . . . . 6 (((z = x w = y) → [w / y][z / x]φ) ↔ (z = x → (w = y → [w / y][z / x]φ)))
75, 6bitri 240 . . . . 5 (((z = x w = y) → [z / x][w / y]φ) ↔ (z = x → (w = y → [w / y][z / x]φ)))
87albii 1566 . . . 4 (w((z = x w = y) → [z / x][w / y]φ) ↔ w(z = x → (w = y → [w / y][z / x]φ)))
9 2sb5rf.2 . . . . . . 7 wφ
109nfsb 2109 . . . . . 6 w[z / x]φ
1110sb6rf 2091 . . . . 5 ([z / x]φw(w = y → [w / y][z / x]φ))
1211imbi2i 303 . . . 4 ((z = x → [z / x]φ) ↔ (z = xw(w = y → [w / y][z / x]φ)))
133, 8, 123bitr4ri 269 . . 3 ((z = x → [z / x]φ) ↔ w((z = x w = y) → [z / x][w / y]φ))
1413albii 1566 . 2 (z(z = x → [z / x]φ) ↔ zw((z = x w = y) → [z / x][w / y]φ))
152, 14bitri 240 1 (φzw((z = x w = y) → [z / x][w / y]φ))
 Colors of variables: wff setvar class Syntax hints:   → 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: (None)
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