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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ceqsalg0 | Structured version Visualization version GIF version |
Description: The FOL content of ceqsalg 3430. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-ceqsalg0.1 | ⊢ Ⅎ𝑥𝜓 |
bj-ceqsalg0.2 | ⊢ (𝜒 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
bj-ceqsalg0 | ⊢ (∃𝑥𝜒 → (∀𝑥(𝜒 → 𝜑) ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-ceqsalg0.1 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | bj-ceqsalg0.2 | . . 3 ⊢ (𝜒 → (𝜑 ↔ 𝜓)) | |
3 | 2 | ax-gen 1802 | . 2 ⊢ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) |
4 | bj-ceqsalt0 34713 | . 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥𝜒) → (∀𝑥(𝜒 → 𝜑) ↔ 𝜓)) | |
5 | 1, 3, 4 | mp3an12 1452 | 1 ⊢ (∃𝑥𝜒 → (∀𝑥(𝜒 → 𝜑) ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1540 ∃wex 1786 Ⅎwnf 1790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-12 2179 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1090 df-ex 1787 df-nf 1791 |
This theorem is referenced by: bj-ceqsalg 34718 bj-ceqsalgv 34720 |
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