Theorem sbie 1802

Description: Conversion of implicit substitution to explicit substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Revised by Wolf Lammen, 30-Apr-2018.)

Hypotheses

Ref	Expression
sbie.1	|- F/ x ps
sbie.2	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
sbie	|- ( [ y / x ] ph <-> ps )

Proof of Theorem sbie

Step	Hyp	Ref	Expression
1		sbie.1	. . 3 |- F/ x ps
2	1	nfri 1530	. 2 |- ( ps -> A. x ps )
3		sbie.2	. 2 |- ( x = y -> ( ph <-> ps ) )
4	2, 3	sbieh 1801	1 |- ( [ y / x ] ph <-> ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1471   [wsb 1773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774

This theorem is referenced by:  sbiev  1803  cbveu  2066  mo4f  2102  bm1.1  2178  eqsb1lem  2296  clelsb1  2298  clelsb2  2299  cbvab  2317  clelsb1f  2340  cbvralf  2718  cbvrexf  2719  cbvreu  2724  sbralie  2744  cbvrab  2758  reu2  2948  rmo4f  2958  nfcdeq  2982  sbcco2  3008  sbcie2g  3019  sbcralt  3062  sbcrext  3063  sbcralg  3064  sbcreug  3066  sbcel12g  3095  sbceqg  3096  cbvralcsf  3143  cbvrexcsf  3144  cbvreucsf  3145  cbvrabcsf  3146  sbss  3554  disjiun  4024  sbcbrg  4083  cbvopab1  4102  cbvmpt  4124  tfis2f  4616  cbviota  5220  relelfvdm  5586  nfvres  5588  cbvriota  5884  bezoutlemnewy  12133  bezoutlemmain  12135