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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 1731 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 1512 | . 2 ⊢ (𝜓 → ∀𝑥𝜓) |
3 | spim.2 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
4 | 2, 3 | spimh 1730 | 1 ⊢ (∀𝑥𝜑 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1346 Ⅎwnf 1453 |
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 1440 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-4 1503 ax-i9 1523 ax-ial 1527 |
This theorem depends on definitions: df-bi 116 df-nf 1454 |
This theorem is referenced by: cbv3 1735 chvar 1750 spimv 1804 2spim 13801 |
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