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	⊢ Ⅎ𝑥𝜓
sbie.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbie	⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

Proof of Theorem sbie

Step	Hyp	Ref	Expression
1		sbie.1	. . 3 ⊢ Ⅎ𝑥𝜓
2	1	nfri 1530	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
3		sbie.2	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	2, 3	sbieh 1801	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 105  Ⅎwnf 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  2949  rmo4f  2959  nfcdeq  2983  sbcco2  3009  sbcie2g  3020  sbcralt  3063  sbcrext  3064  sbcralg  3065  sbcreug  3067  sbcel12g  3096  sbceqg  3097  cbvralcsf  3144  cbvrexcsf  3145  cbvreucsf  3146  cbvrabcsf  3147  sbss  3555  disjiun  4025  sbcbrg  4084  cbvopab1  4103  cbvmpt  4125  tfis2f  4617  cbviota  5221  relelfvdm  5587  nfvres  5589  cbvriota  5885  bezoutlemnewy  12136  bezoutlemmain  12138