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Theorem sbex 1877
 Description: Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.)
Assertion
Ref Expression
sbex ([z / y]xφx[z / y]φ)
Distinct variable groups:   x,y   x,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem sbex
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 sbexyz 1876 . . . 4 ([w / y]xφx[w / y]φ)
21sbbii 1645 . . 3 ([z / w][w / y]xφ ↔ [z / w]x[w / y]φ)
3 sbexyz 1876 . . 3 ([z / w]x[w / y]φx[z / w][w / y]φ)
42, 3bitri 173 . 2 ([z / w][w / y]xφx[z / w][w / y]φ)
5 ax-17 1416 . . 3 (xφwxφ)
65sbco2v 1818 . 2 ([z / w][w / y]xφ ↔ [z / y]xφ)
7 ax-17 1416 . . . 4 (φwφ)
87sbco2v 1818 . . 3 ([z / w][w / y]φ ↔ [z / y]φ)
98exbii 1493 . 2 (x[z / w][w / y]φx[z / y]φ)
104, 6, 93bitr3i 199 1 ([z / y]xφx[z / y]φ)
 Colors of variables: wff set class Syntax hints:   ↔ 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-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:  sbabel  2200  sbcex2  2806  sbcexg  2807
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