Theorem bj-equsalvwd 37429

Description: Variant of equsalvw 2037. (Contributed by BJ, 7-Oct-2024.)

Hypotheses

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

Assertion

Ref	Expression
bj-equsalvwd	⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜒))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem bj-equsalvwd

Step	Hyp	Ref	Expression
1		bj-equsalvwd.nf0	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2		bj-equsalvwd.is	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
3	2	pm5.74da 816	. . 3 ⊢ (𝜑 → ((𝑥 = 𝑦 → 𝜓) ↔ (𝑥 = 𝑦 → 𝜒)))
4	1, 3	albidh 1899	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜒)))
5		bj-equsalvwd.nf	. . 3 ⊢ (𝜑 → Ⅎ'𝑥𝜒)
6		bj-equsvt 37428	. . 3 ⊢ (Ⅎ'𝑥𝜒 → (∀𝑥(𝑥 = 𝑦 → 𝜒) ↔ 𝜒))
7	5, 6	syl 18	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜒) ↔ 𝜒))
8	4, 7	bitrd 282	1 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  Ⅎ'wnnf 37383

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-6 2000

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-bj-nnf 37384

This theorem is used by:  bj-equsexvwd  37430  bj-sbievwd  37434