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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-spimedv | Structured version Visualization version GIF version |
Description: Version of spimed 2417 with a dv condition, which does not require ax-13 2408. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-spimedv.1 | ⊢ (𝜒 → Ⅎ𝑥𝜑) |
bj-spimedv.2 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
bj-spimedv | ⊢ (𝜒 → (𝜑 → ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-spimedv.1 | . . 3 ⊢ (𝜒 → Ⅎ𝑥𝜑) | |
2 | 1 | nf5rd 2220 | . 2 ⊢ (𝜒 → (𝜑 → ∀𝑥𝜑)) |
3 | ax6ev 2059 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝑦 | |
4 | bj-spimedv.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
5 | 3, 4 | eximii 1912 | . . 3 ⊢ ∃𝑥(𝜑 → 𝜓) |
6 | 5 | 19.35i 1958 | . 2 ⊢ (∀𝑥𝜑 → ∃𝑥𝜓) |
7 | 2, 6 | syl6 35 | 1 ⊢ (𝜒 → (𝜑 → ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1629 ∃wex 1852 Ⅎwnf 1856 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-12 2203 |
This theorem depends on definitions: df-bi 197 df-ex 1853 df-nf 1858 |
This theorem is referenced by: bj-spimev 33057 |
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