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Mirrors > Home > ILE Home > Th. List > Mathboxes > spimd | GIF version |
Description: Deduction form of spim 1731. (Contributed by BJ, 17-Oct-2019.) |
Ref | Expression |
---|---|
spimd.nf | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
spimd.1 | ⊢ (𝜑 → ∀𝑥(𝑥 = 𝑦 → (𝜓 → 𝜒))) |
Ref | Expression |
---|---|
spimd | ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spimd.nf | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
2 | spimd.1 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝑦 → (𝜓 → 𝜒))) | |
3 | spimt 1729 | . 2 ⊢ ((Ⅎ𝑥𝜒 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜓 → 𝜒))) → (∀𝑥𝜓 → 𝜒)) | |
4 | 1, 2, 3 | syl2anc 409 | 1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) |
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: 2spim 13801 |
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