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Theorem 2sb5 1856
 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 1764 . 2 ([z / x][w / y]φx(x = z [w / y]φ))
2 19.42v 1783 . . . 4 (y(x = z (y = w φ)) ↔ (x = z y(y = w φ)))
3 anass 381 . . . . 5 (((x = z y = w) φ) ↔ (x = z (y = w φ)))
43exbii 1493 . . . 4 (y((x = z y = w) φ) ↔ y(x = z (y = w φ)))
5 sb5 1764 . . . . 5 ([w / y]φy(y = w φ))
65anbi2i 430 . . . 4 ((x = z [w / y]φ) ↔ (x = z y(y = w φ)))
72, 4, 63bitr4ri 202 . . 3 ((x = z [w / y]φ) ↔ y((x = z y = w) φ))
87exbii 1493 . 2 (x(x = z [w / y]φ) ↔ xy((x = z y = w) φ))
91, 8bitri 173 1 ([z / x][w / y]φxy((x = z y = w) φ))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∃wex 1378  [wsb 1642 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424 This theorem depends on definitions:  df-bi 110  df-sb 1643 This theorem is referenced by:  opelopabsbALT  3987
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