Theorem sbie 2506

Description: Conversion of implicit substitution to explicit substitution. For versions requiring disjoint variables, but fewer axioms, see sbiev 2312 and sbievw 2097. 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 2078	. . 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 2266	. . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜓)
7	6	sblbis 2309	. 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 1787  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-12 2173  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-nf 1788  df-sb 2069

This theorem is referenced by:  sbied  2507  2sbiev  2509  cbvmo  2605  cbveu  2609  cbvab  2815  clelsb2  2867  cbvralf  3361  cbvreu  3370  cbvrab  3415  nfcdeq  3707  cbvralcsf  3873  cbvreucsf  3875  cbvrabcsf  3876  cbvopab1g  5146  cbvmptfg  5180  cbviota  6386  cbvriota  7226  nd1  10274  nd2  10275  sbcrexgOLD  40523