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Theorem 2sb5 2112
 Description: Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
Assertion
Ref Expression
2sb5 ([z / x][w / y]φxy((x = z y = w) φ))
Distinct variable groups:   x,y,z   y,w
Allowed substitution hints:   φ(x,y,z,w)

Proof of Theorem 2sb5
StepHypRef Expression
1 sb5 2100 . 2 ([z / x][w / y]φx(x = z [w / y]φ))
2 19.42v 1905 . . . 4 (y(x = z (y = w φ)) ↔ (x = z y(y = w φ)))
3 anass 630 . . . . 5 (((x = z y = w) φ) ↔ (x = z (y = w φ)))
43exbii 1582 . . . 4 (y((x = z y = w) φ) ↔ y(x = z (y = w φ)))
5 sb5 2100 . . . . 5 ([w / y]φy(y = w φ))
65anbi2i 675 . . . 4 ((x = z [w / y]φ) ↔ (x = z y(y = w φ)))
72, 4, 63bitr4ri 269 . . 3 ((x = z [w / y]φ) ↔ y((x = z y = w) φ))
87exbii 1582 . 2 (x(x = z [w / y]φ) ↔ xy((x = z y = w) φ))
91, 8bitri 240 1 ([z / x][w / y]φxy((x = z y = w) φ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541  [wsb 1648 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 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:  pm11.07  2115
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