Theorem bj-sbievv 34191

Description: Version of sbie 2543 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 2536	. 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 2438	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
6	2, 5	bitri 277	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1534  Ⅎwnf 1783  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-12 2176  ax-13 2389

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1780  df-nf 1784  df-sb 2069

This theorem is referenced by: (None)