Theorem sbie 2534

Description: Conversion of implicit substitution to explicit substitution. For versions requiring disjoint variables, but fewer axioms, see sbiev 2348 and sbievw 2132. Usage of this theorem is discouraged because it depends on ax-13 2404. (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 2523	. . 3 ⊢ [𝑦 / 𝑥]𝑥 = 𝑦
2		sbie.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	sbimi 2112	. . 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 2307	. . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜓)
7	6	sblbis 2344	. 2 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝜓))
8	4, 7	mpbi 233	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  Ⅎwnf 1815  [wsb 2100

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-10 2179  ax-12 2215  ax-13 2404

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-nf 1816  df-sb 2101

This theorem is used by:  sbied  2535  2sbiev  2537  cbvmo  2632  cbveu  2635  cbvab  2836  cbvralf  3349  cbvreu  3408  cbvrab  3453  nfcdeq  3738  cbvralcsf  3893  cbvreucsf  3895  cbvrabcsf  3896  cbvopab1g  5183  cbvmptfg  5209  cbviota  6510  cbvriota  7400  nd1  10685  nd2  10686