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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-3exbi | Structured version Visualization version GIF version |
Description: Closed form of 3exbii 1853. (Contributed by BJ, 6-May-2019.) |
Ref | Expression |
---|---|
bj-3exbi | ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exbi 1850 | . . 3 ⊢ (∀𝑧(𝜑 ↔ 𝜓) → (∃𝑧𝜑 ↔ ∃𝑧𝜓)) | |
2 | 1 | 2alimi 1816 | . 2 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) → ∀𝑥∀𝑦(∃𝑧𝜑 ↔ ∃𝑧𝜓)) |
3 | bj-2exbi 34766 | . 2 ⊢ (∀𝑥∀𝑦(∃𝑧𝜑 ↔ ∃𝑧𝜓) → (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓)) | |
4 | 2, 3 | syl 17 | 1 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1537 ∃wex 1783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 |
This theorem depends on definitions: df-bi 206 df-ex 1784 |
This theorem is referenced by: (None) |
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