Mathbox for Emmett Weisz |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > spd | Structured version Visualization version GIF version |
Description: Specialization deduction, using implicit substitution. Based on the proof of spimed 2243. (Contributed by Emmett Weisz, 17-Jan-2020.) |
Ref | Expression |
---|---|
spd.1 | ⊢ (𝜒 → Ⅎ𝑥𝜓) |
spd.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
spd | ⊢ (𝜒 → (∀𝑥𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6e 2238 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | spd.2 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | biimpd 218 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
4 | 1, 3 | eximii 1754 | . . 3 ⊢ ∃𝑥(𝜑 → 𝜓) |
5 | 4 | 19.35i 1795 | . 2 ⊢ (∀𝑥𝜑 → ∃𝑥𝜓) |
6 | spd.1 | . . 3 ⊢ (𝜒 → Ⅎ𝑥𝜓) | |
7 | 6 | 19.9d 2058 | . 2 ⊢ (𝜒 → (∃𝑥𝜓 → 𝜓)) |
8 | 5, 7 | syl5 33 | 1 ⊢ (𝜒 → (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 ∃wex 1695 Ⅎwnf 1699 |
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-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-nf 1701 |
This theorem is referenced by: (None) |
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