Theorem sb5f 2386

Description: Equivalence for substitution when y is not free in ph. The implication "to the right" is sb1 1883 and does not require the non-freeness hypothesis. Theorem sb5 2430 replaces the non-freeness hypothesis with a dv condition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Hypothesis

Ref	Expression
sb6f.1	|- F/ y ph

Assertion

Ref	Expression
sb5f	|- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )

Proof of Theorem sb5f

Step	Hyp	Ref	Expression
1		sb6f.1	. . 3 |- F/ y ph
2	1	sb6f 2385	. 2 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
3	1	equs45f 2350	. 2 |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )
4	2, 3	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   F/wnf 1708   [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:  sb7f  2453