Theorem bj-sbievwd 34891

Description: Variant of sbievw 2097. (Contributed by BJ, 7-Oct-2024.)

Hypotheses

Ref	Expression
bj-sbievwd.nf0	⊢ (𝜑 → ∀𝑥𝜑)
bj-sbievwd.nf	⊢ (𝜑 → Ⅎ'𝑥𝜒)
bj-sbievwd.is	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
bj-sbievwd	⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem bj-sbievwd

Step	Hyp	Ref	Expression
1		sb6 2089	. 2 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓))
2		bj-sbievwd.nf0	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3		bj-sbievwd.nf	. . 3 ⊢ (𝜑 → Ⅎ'𝑥𝜒)
4		bj-sbievwd.is	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
5	2, 3, 4	bj-equsalvwd 34889	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜒))
6	1, 5	syl5bb 282	1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537  [wsb 2068  Ⅎ'wnnf 34832

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

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-sb 2069  df-bj-nnf 34833

This theorem is referenced by: (None)