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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-equsexvwd | Structured version Visualization version GIF version |
Description: Variant of equsexvw 2014. (Contributed by BJ, 7-Oct-2024.) |
Ref | Expression |
---|---|
bj-equsexvwd.nf0 | ⊢ (𝜑 → ∀𝑥𝜑) |
bj-equsexvwd.nf | ⊢ (𝜑 → Ⅎ'𝑥𝜒) |
bj-equsexvwd.is | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
bj-equsexvwd | ⊢ (𝜑 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ 𝜒)) |
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)) |
Colors of variables: wff setvar class |
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) |
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