Theorem bj-sbievv 34287

Description: Version of sbie 2521 with a second nonfreeness hypothesis and shorter proof. (Contributed by BJ, 18-Jul-2023.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-sbievv.nfx	⊢ Ⅎ𝑥𝜓
bj-sbievv.nfy	⊢ Ⅎ𝑦𝜑
bj-sbievv.is	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
bj-sbievv	⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

Proof of Theorem bj-sbievv

Step	Hyp	Ref	Expression
1		bj-sbievv.nfy	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	sb6f 2515	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
3		bj-sbievv.nfx	. . 3 ⊢ Ⅎ𝑥𝜓
4		bj-sbievv.is	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	3, 4	equsal 2428	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
6	2, 5	bitri 278	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  Ⅎwnf 1785  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175  ax-13 2379

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070

This theorem is referenced by: (None)