Theorem sbie 2506

Description: Conversion of implicit substitution to explicit substitution. For versions requiring disjoint variables, but fewer axioms, see sbiev 2309 and sbievw 2095. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 13-Jul-2019.) (New usage is discouraged.)

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		equsb1 2495	. . 3 ⊢ [𝑦 / 𝑥]𝑥 = 𝑦
2		sbie.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	sbimi 2077	. . 3 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑦 → [𝑦 / 𝑥](𝜑 ↔ 𝜓))
4	1, 3	ax-mp 5	. 2 ⊢ [𝑦 / 𝑥](𝜑 ↔ 𝜓)
5		sbie.1	. . . 4 ⊢ Ⅎ𝑥𝜓
6	5	sbf 2263	. . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜓)
7	6	sblbis 2306	. 2 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝜓))
8	4, 7	mpbi 229	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 205  Ⅎwnf 1786  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-nf 1787  df-sb 2068

This theorem is referenced by:  sbied  2507  2sbiev  2509  cbvmo  2605  cbveu  2609  cbvab  2814  clelsb2OLD  2868  cbvralf  3371  cbvreu  3381  cbvrab  3425  nfcdeq  3712  cbvralcsf  3877  cbvreucsf  3879  cbvrabcsf  3880  cbvopab1g  5150  cbvmptfg  5184  cbviota  6401  cbvriota  7246  nd1  10343  nd2  10344  sbcrexgOLD  40607