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Mirrors > Home > ILE Home > Th. List > Mathboxes > 2spim | GIF version |
Description: Double substitution, as in spim 1749. (Contributed by BJ, 17-Oct-2019.) |
Ref | Expression |
---|---|
2spim.nfx | ⊢ Ⅎ𝑥𝜒 |
2spim.nfz | ⊢ Ⅎ𝑧𝜒 |
2spim.1 | ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
2spim | ⊢ (∀𝑧∀𝑥𝜓 → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2spim.nfz | . 2 ⊢ Ⅎ𝑧𝜒 | |
2 | 2spim.nfx | . . . 4 ⊢ Ⅎ𝑥𝜒 | |
3 | 2 | a1i 9 | . . 3 ⊢ (𝑧 = 𝑡 → Ⅎ𝑥𝜒) |
4 | 2spim.1 | . . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜓 → 𝜒)) | |
5 | 4 | expcom 116 | . . . 4 ⊢ (𝑧 = 𝑡 → (𝑥 = 𝑦 → (𝜓 → 𝜒))) |
6 | 5 | alrimiv 1885 | . . 3 ⊢ (𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → (𝜓 → 𝜒))) |
7 | 3, 6 | spimd 14975 | . 2 ⊢ (𝑧 = 𝑡 → (∀𝑥𝜓 → 𝜒)) |
8 | 1, 7 | spim 1749 | 1 ⊢ (∀𝑧∀𝑥𝜓 → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∀wal 1362 Ⅎwnf 1471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 |
This theorem depends on definitions: df-bi 117 df-nf 1472 |
This theorem is referenced by: ch2var 14977 |
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