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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-sbimedh | GIF version |
Description: A strengthening of sbiedh 1710 (same proof). (Contributed by BJ, 16-Dec-2019.) |
Ref | Expression |
---|---|
bj-sbimedh.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
bj-sbimedh.2 | ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) |
bj-sbimedh.3 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒))) |
Ref | Expression |
---|---|
bj-sbimedh | ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb1 1689 | . . 3 ⊢ ([𝑦 / 𝑥]𝜓 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)) | |
2 | bj-sbimedh.1 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | bj-sbimedh.3 | . . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒))) | |
4 | 3 | impd 251 | . . . 4 ⊢ (𝜑 → ((𝑥 = 𝑦 ∧ 𝜓) → 𝜒)) |
5 | 2, 4 | eximdh 1542 | . . 3 ⊢ (𝜑 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) → ∃𝑥𝜒)) |
6 | 1, 5 | syl5 32 | . 2 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → ∃𝑥𝜒)) |
7 | bj-sbimedh.2 | . . 3 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) | |
8 | 2, 7 | 19.9hd 1592 | . 2 ⊢ (𝜑 → (∃𝑥𝜒 → 𝜒)) |
9 | 6, 8 | syld 44 | 1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∀wal 1282 ∃wex 1421 [wsb 1685 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-ial 1467 |
This theorem depends on definitions: df-bi 115 df-sb 1686 |
This theorem is referenced by: bj-sbimeh 10583 |
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