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Theorem 2sb5rf 1862
 Description: Reversed double substitution. (Contributed by NM, 3-Feb-2005.)
Hypotheses
Ref Expression
2sb5rf.1 (φzφ)
2sb5rf.2 (φwφ)
Assertion
Ref Expression
2sb5rf (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 2sb5rf
StepHypRef Expression
1 2sb5rf.1 . . 3 (φzφ)
21sb5rf 1729 . 2 (φz(z = x [z / x]φ))
3 19.42v 1783 . . . 4 (w(z = x (w = y [w / y][z / x]φ)) ↔ (z = x w(w = y [w / y][z / x]φ)))
4 sbcom2 1860 . . . . . . 7 ([z / x][w / y]φ ↔ [w / y][z / x]φ)
54anbi2i 430 . . . . . 6 (((z = x w = y) [z / x][w / y]φ) ↔ ((z = x w = y) [w / y][z / x]φ))
6 anass 381 . . . . . 6 (((z = x w = y) [w / y][z / x]φ) ↔ (z = x (w = y [w / y][z / x]φ)))
75, 6bitri 173 . . . . 5 (((z = x w = y) [z / x][w / y]φ) ↔ (z = x (w = y [w / y][z / x]φ)))
87exbii 1493 . . . 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 (φwφ)
109hbsbv 1814 . . . . . 6 ([z / x]φw[z / x]φ)
1110sb5rf 1729 . . . . 5 ([z / x]φw(w = y [w / y][z / x]φ))
1211anbi2i 430 . . . 4 ((z = x [z / x]φ) ↔ (z = x w(w = y [w / y][z / x]φ)))
133, 8, 123bitr4ri 202 . . 3 ((z = x [z / x]φ) ↔ w((z = x w = y) [z / x][w / y]φ))
1413exbii 1493 . 2 (z(z = x [z / x]φ) ↔ zw((z = x w = y) [z / x][w / y]φ))
152, 14bitri 173 1 (φzw((z = x w = y) [z / x][w / y]φ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240  ∃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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643 This theorem is referenced by: (None)
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