Theorem sb6f 2385

Description: Equivalence for substitution when y is not free in ph. The implication "to the left" is sb2 2352 and does not require the non-freeness hypothesis. Theorem sb6 2429 replaces the non-freeness hypothesis with a dv condition. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Hypothesis

Ref	Expression
sb6f.1	|- F/ y ph

Assertion

Ref	Expression
sb6f	|- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )

Proof of Theorem sb6f

Step	Hyp	Ref	Expression
1		sb6f.1	. . . . 5 |- F/ y ph
2	1	nf5ri 2065	. . . 4 |- ( ph -> A. y ph )
3	2	sbimi 1886	. . 3 |- ( [ y / x ] ph -> [ y / x ] A. y ph )
4		sb4a 2357	. . 3 |- ( [ y / x ] A. y ph -> A. x ( x = y -> ph ) )
5	3, 4	syl 17	. 2 |- ( [ y / x ] ph -> A. x ( x = y -> ph ) )
6		sb2 2352	. 2 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
7	5, 6	impbii 199	1 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   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:  sb5f  2386