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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-sbimeh | GIF version |
Description: A strengthening of sbieh 1770 (same proof). (Contributed by BJ, 16-Dec-2019.) |
Ref | Expression |
---|---|
bj-sbimeh.1 | ⊢ (𝜓 → ∀𝑥𝜓) |
bj-sbimeh.2 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
bj-sbimeh | ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tru 1339 | . . . 4 ⊢ ⊤ | |
2 | 1 | hbth 1443 | . . 3 ⊢ (⊤ → ∀𝑥⊤) |
3 | bj-sbimeh.1 | . . . 4 ⊢ (𝜓 → ∀𝑥𝜓) | |
4 | 3 | a1i 9 | . . 3 ⊢ (⊤ → (𝜓 → ∀𝑥𝜓)) |
5 | bj-sbimeh.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
6 | 5 | a1i 9 | . . 3 ⊢ (⊤ → (𝑥 = 𝑦 → (𝜑 → 𝜓))) |
7 | 2, 4, 6 | bj-sbimedh 13356 | . 2 ⊢ (⊤ → ([𝑦 / 𝑥]𝜑 → 𝜓)) |
8 | 7 | mptru 1344 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1333 ⊤wtru 1336 [wsb 1742 |
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-ial 1514 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-sb 1743 |
This theorem is referenced by: bj-sbime 13358 |
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