Theorem bj-sbievwd 34650

Description: Variant of sbievw 2101. (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 2093	. 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 34648	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜒))
6	1, 5	syl5bb 286	1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1541  [wsb 2072  Ⅎ'wnnf 34591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-sb 2073  df-bj-nnf 34592

This theorem is referenced by: (None)