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Mirrors > Home > MPE Home > Th. List > Mathboxes > 19.41rg | Structured version Visualization version GIF version |
Description: Closed form of right-to-left implication of 19.41 2226, Theorem 19.41 of [Margaris] p. 90. Derived from 19.41rgVD 42559. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
19.41rg | ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sp 2174 | . . . 4 ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (𝜓 → ∀𝑥𝜓)) | |
2 | pm3.21 473 | . . . . . . 7 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓))) | |
3 | 2 | a1i 11 | . . . . . 6 ⊢ ((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → (𝜑 ∧ 𝜓)))) |
4 | 3 | al2imi 1815 | . . . . 5 ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑 ∧ 𝜓)))) |
5 | exim 1834 | . . . . 5 ⊢ (∀𝑥(𝜑 → (𝜑 ∧ 𝜓)) → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))) | |
6 | 4, 5 | syl6 35 | . . . 4 ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))) |
7 | 1, 6 | syld 47 | . . 3 ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))) |
8 | 7 | com23 86 | . 2 ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑 ∧ 𝜓)))) |
9 | 8 | impd 412 | 1 ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∀wal 1537 ∃wex 1779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-12 2169 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1780 |
This theorem is referenced by: ax6e2nd 42215 ax6e2ndVD 42565 ax6e2ndALT 42587 |
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