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Mirrors > Home > MPE Home > Th. List > sbequ2OLD | Structured version Visualization version GIF version |
Description: Obsolete version of sbequ2 2250 as of 3-Feb-2024. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Feb-2018.) Revise df-sb 2070. (Revised by BJ, 22-Dec-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
sbequ2OLD | ⊢ (𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equvinva 2037 | . 2 ⊢ (𝑥 = 𝑡 → ∃𝑦(𝑥 = 𝑦 ∧ 𝑡 = 𝑦)) | |
2 | df-sb 2070 | . . . 4 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
3 | equcomi 2024 | . . . . . . 7 ⊢ (𝑡 = 𝑦 → 𝑦 = 𝑡) | |
4 | sp 2182 | . . . . . . 7 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑)) | |
5 | 3, 4 | imim12i 62 | . . . . . 6 ⊢ ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → (𝑡 = 𝑦 → (𝑥 = 𝑦 → 𝜑))) |
6 | 5 | impcomd 414 | . . . . 5 ⊢ ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ((𝑥 = 𝑦 ∧ 𝑡 = 𝑦) → 𝜑)) |
7 | 6 | alimi 1812 | . . . 4 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∀𝑦((𝑥 = 𝑦 ∧ 𝑡 = 𝑦) → 𝜑)) |
8 | 2, 7 | sylbi 219 | . . 3 ⊢ ([𝑡 / 𝑥]𝜑 → ∀𝑦((𝑥 = 𝑦 ∧ 𝑡 = 𝑦) → 𝜑)) |
9 | 19.23v 1943 | . . 3 ⊢ (∀𝑦((𝑥 = 𝑦 ∧ 𝑡 = 𝑦) → 𝜑) ↔ (∃𝑦(𝑥 = 𝑦 ∧ 𝑡 = 𝑦) → 𝜑)) | |
10 | 8, 9 | sylib 220 | . 2 ⊢ ([𝑡 / 𝑥]𝜑 → (∃𝑦(𝑥 = 𝑦 ∧ 𝑡 = 𝑦) → 𝜑)) |
11 | 1, 10 | syl5com 31 | 1 ⊢ (𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1535 ∃wex 1780 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 |
This theorem is referenced by: (None) |
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