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Mirrors > Home > ILE Home > Th. List > spimeh | GIF version |
Description: Existential introduction, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by NM, 3-Feb-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
spimeh.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
spimeh.2 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
spimeh | ⊢ (𝜑 → ∃𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | a9e 1586 | . 2 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | spimeh.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | spimeh.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
4 | 3 | com12 27 | . . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → 𝜓)) |
5 | 2, 4 | eximdh 1502 | . 2 ⊢ (𝜑 → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜓)) |
6 | 1, 5 | mpi 15 | 1 ⊢ (𝜑 → ∃𝑥𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1241 = wceq 1243 ∃wex 1381 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-i9 1423 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: (None) |
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