Theorem sb5 1645

Description: Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40.

Assertion

Ref	Expression
sb5	|- ([y / x]ph <-> E.x(x = y /\ ph))

Distinct variable group:   x,y

Proof of Theorem sb5

Step	Hyp	Ref	Expression
1		sb6 1644	. 2 |- ([y / x]ph <-> A.x(x = y -> ph))
2		sb56 1643	. 2 |- (E.x(x = y /\ ph) <-> A.x(x = y -> ph))
3	1, 2	bitr4i 193	1 |- ([y / x]ph <-> E.x(x = y /\ ph))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298  E.wex 1326  [wsbc 1534

This theorem is referenced by:  2sb5 1725  dfsb7 1730  sb7f 1731  sbelx 1735  sbc5g 2470  pm14.122b 16387

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-16 1580  ax-11o 1588

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536

Copyright terms: Public domain