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Mirrors > Home > ILE Home > Th. List > spimh | GIF version |
Description: Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The spim 1718 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 8-May-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
spimh.1 | ⊢ (𝜓 → ∀𝑥𝜓) |
spimh.2 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
spimh | ⊢ (∀𝑥𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spimh.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
2 | spimh.1 | . . . 4 ⊢ (𝜓 → ∀𝑥𝜓) | |
3 | 1, 2 | syl6com 35 | . . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → ∀𝑥𝜓)) |
4 | 3 | alimi 1435 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓)) |
5 | ax9o 1678 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓) → 𝜓) | |
6 | 4, 5 | syl 14 | 1 ⊢ (∀𝑥𝜑 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1333 |
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 1427 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-4 1490 ax-i9 1510 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: spim 1718 |
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