Theorem bj-equsexvwd 34649

Description: Variant of equsexvw 2014. (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 1850	. . 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 34608	. . . . 5 ⊢ (Ⅎ'𝑥𝜒 ↔ Ⅎ'𝑥 ¬ 𝜒)
5	3, 4	sylib 221	. . . 4 ⊢ (𝜑 → Ⅎ'𝑥 ¬ 𝜒)
6		bj-equsexvwd.is	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
7	6	notbid 321	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (¬ 𝜓 ↔ ¬ 𝜒))
8	2, 5, 7	bj-equsalvwd 34648	. . 3 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → ¬ 𝜓) ↔ ¬ 𝜒))
9	1, 8	bitr3id 288	. 2 ⊢ (𝜑 → (¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ ¬ 𝜒))
10	9	con4bid 320	1 ⊢ (𝜑 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1541  ∃wex 1787  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-6 1976

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

This theorem is referenced by: (None)