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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-spimtv | Structured version Visualization version GIF version |
Description: Version of spimt 2393 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-spimtv | ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6ev 2074 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | exim 1929 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝜑 → 𝜓))) | |
3 | 1, 2 | mpi 20 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ∃𝑥(𝜑 → 𝜓)) |
4 | 19.35 1977 | . . 3 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) | |
5 | 3, 4 | sylib 210 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → ∃𝑥𝜓)) |
6 | 19.9t 2238 | . . 3 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 ↔ 𝜓)) | |
7 | 6 | biimpd 221 | . 2 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 → 𝜓)) |
8 | 5, 7 | sylan9r 505 | 1 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∀wal 1651 ∃wex 1875 Ⅎwnf 1879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-12 2213 |
This theorem depends on definitions: df-bi 199 df-an 386 df-ex 1876 df-nf 1880 |
This theorem is referenced by: (None) |
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