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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-19.41al | Structured version Visualization version GIF version |
Description: Special case of 19.41 2090 proved from Tarski, ax-10 2006 (modal5) and hba1 2137 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-19.41al | ⊢ (∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.40 1785 | . . 3 ⊢ (∃𝑥(𝜑 ∧ ∀𝑥𝜓) → (∃𝑥𝜑 ∧ ∃𝑥∀𝑥𝜓)) | |
2 | bj-modal5e 31825 | . . . 4 ⊢ (∃𝑥∀𝑥𝜓 → ∀𝑥𝜓) | |
3 | 2 | anim2i 591 | . . 3 ⊢ ((∃𝑥𝜑 ∧ ∃𝑥∀𝑥𝜓) → (∃𝑥𝜑 ∧ ∀𝑥𝜓)) |
4 | 1, 3 | syl 17 | . 2 ⊢ (∃𝑥(𝜑 ∧ ∀𝑥𝜓) → (∃𝑥𝜑 ∧ ∀𝑥𝜓)) |
5 | hba1 2137 | . . . 4 ⊢ (∀𝑥𝜓 → ∀𝑥∀𝑥𝜓) | |
6 | 5 | anim2i 591 | . . 3 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → (∃𝑥𝜑 ∧ ∀𝑥∀𝑥𝜓)) |
7 | 19.29r 1790 | . . 3 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥∀𝑥𝜓) → ∃𝑥(𝜑 ∧ ∀𝑥𝜓)) | |
8 | 6, 7 | syl 17 | . 2 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ ∀𝑥𝜓)) |
9 | 4, 8 | impbii 198 | 1 ⊢ (∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∀wal 1473 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ex 1696 df-nf 1701 |
This theorem is referenced by: bj-equsexval 31827 |
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