Theorem bj-equsexvwd 34890

Description: Variant of equsexvw 2009. (Contributed by BJ, 7-Oct-2024.)

Hypotheses

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

Assertion

Ref	Expression
bj-equsexvwd	⊢ (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-equsexvwd

Step	Hyp	Ref	Expression
1		alinexa 1846	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))
2		bj-equsexvwd.nf0	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
3		bj-equsexvwd.nf	. . . . 5 ⊢ (𝜑 → Ⅎ'𝑥𝜒)
4		bj-nnfnt 34849	. . . . 5 ⊢ (Ⅎ'𝑥𝜒 ↔ Ⅎ'𝑥 ¬ 𝜒)
5	3, 4	sylib 217	. . . 4 ⊢ (𝜑 → Ⅎ'𝑥 ¬ 𝜒)
6		bj-equsexvwd.is	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
7	6	notbid 317	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (¬ 𝜓 ↔ ¬ 𝜒))
8	2, 5, 7	bj-equsalvwd 34889	. . 3 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → ¬ 𝜓) ↔ ¬ 𝜒))
9	1, 8	bitr3id 284	. 2 ⊢ (𝜑 → (¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ ¬ 𝜒))
10	9	con4bid 316	1 ⊢ (𝜑 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537  ∃wex 1783  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-6 1972

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

This theorem is referenced by: (None)