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Mirrors > Home > MPE Home > Th. List > ex-natded9.26-2 | Structured version Visualization version GIF version |
Description: A more efficient proof of Theorem 9.26 of [Clemente] p. 45. Compare with ex-natded9.26 28771. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ex-natded9.26.1 | ⊢ (𝜑 → ∃𝑥∀𝑦𝜓) |
Ref | Expression |
---|---|
ex-natded9.26-2 | ⊢ (𝜑 → ∀𝑦∃𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ex-natded9.26.1 | . . 3 ⊢ (𝜑 → ∃𝑥∀𝑦𝜓) | |
2 | sp 2180 | . . . 4 ⊢ (∀𝑦𝜓 → 𝜓) | |
3 | 2 | eximi 1841 | . . 3 ⊢ (∃𝑥∀𝑦𝜓 → ∃𝑥𝜓) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝜑 → ∃𝑥𝜓) |
5 | 4 | alrimiv 1934 | 1 ⊢ (𝜑 → ∀𝑦∃𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 ∃wex 1786 |
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 2015 ax-12 2175 |
This theorem depends on definitions: df-bi 206 df-ex 1787 |
This theorem is referenced by: (None) |
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