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Mirrors > Home > ILE Home > Th. List > spim | GIF version |
Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1716 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.) |
Ref | Expression |
---|---|
spim.1 | ⊢ Ⅎ𝑥𝜓 |
spim.2 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
spim | ⊢ (∀𝑥𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spim.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | 1 | nfri 1499 | . 2 ⊢ (𝜓 → ∀𝑥𝜓) |
3 | spim.2 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
4 | 2, 3 | spimh 1715 | 1 ⊢ (∀𝑥𝜑 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1329 Ⅎwnf 1436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-i9 1510 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-nf 1437 |
This theorem is referenced by: cbv3 1720 chvar 1730 spimv 1783 2spim 12973 |
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