Theorem sbie 1837

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 1565	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
3		sbie.2	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	2, 3	sbieh 1836	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 105  Ⅎwnf 1506  [wsb 1808

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-i9 1576  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809

This theorem is referenced by:  sbiev  1838  cbveu  2101  mo4f  2138  bm1.1  2214  eqsb1lem  2332  clelsb1  2334  clelsb2  2335  cbvab  2353  clelsb1f  2376  cbvralf  2756  cbvrexf  2757  cbvreu  2763  sbralie  2783  cbvrab  2798  reu2  2992  rmo4f  3002  nfcdeq  3026  sbcco2  3052  sbcie2g  3063  sbcralt  3106  sbcrext  3107  sbcralg  3108  sbcreug  3110  sbcel12g  3140  sbceqg  3141  cbvralcsf  3188  cbvrexcsf  3189  cbvreucsf  3190  cbvrabcsf  3191  sbss  3600  disjiun  4081  sbcbrg  4141  cbvopab1  4160  cbvmpt  4182  tfis2f  4680  cbviota  5289  relelfvdm  5667  nfvres  5671  cbvriota  5978  bezoutlemnewy  12557  bezoutlemmain  12559