Theorem sb5 2430

Description: Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. The implication "to the right" is sb1 1883 and does not require any dv condition. Theorem sb5f 2386 replaces the dv condition with a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.)

Assertion

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

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem sb5

Step	Hyp	Ref	Expression
1		sb6 2429	. 2 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
2		sb56 2150	. 2 |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )
3	1, 2	bitr4i 267	1 |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  2sb5  2443  sb7f  2453  sbelx  2458  sbc2or  3444  sbc5  3460