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Mirrors > Home > MPE Home > Th. List > sbequ2 | Structured version Visualization version GIF version |
Description: An equality theorem for substitution. (Contributed by NM, 16-May-1993.) Revise df-sb 2066. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Wolf Lammen, 3-Feb-2024.) |
Ref | Expression |
---|---|
sbequ2 | ⊢ (𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sb 2066 | . . . 4 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
2 | 1 | biimpi 218 | . . 3 ⊢ ([𝑡 / 𝑥]𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
3 | equvinva 2033 | . . 3 ⊢ (𝑥 = 𝑡 → ∃𝑦(𝑥 = 𝑦 ∧ 𝑡 = 𝑦)) | |
4 | equcomi 2020 | . . . . . 6 ⊢ (𝑡 = 𝑦 → 𝑦 = 𝑡) | |
5 | sp 2178 | . . . . . 6 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑)) | |
6 | 4, 5 | imim12i 62 | . . . . 5 ⊢ ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → (𝑡 = 𝑦 → (𝑥 = 𝑦 → 𝜑))) |
7 | 6 | impcomd 414 | . . . 4 ⊢ ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ((𝑥 = 𝑦 ∧ 𝑡 = 𝑦) → 𝜑)) |
8 | 7 | aleximi 1828 | . . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → (∃𝑦(𝑥 = 𝑦 ∧ 𝑡 = 𝑦) → ∃𝑦𝜑)) |
9 | 2, 3, 8 | syl2im 40 | . 2 ⊢ ([𝑡 / 𝑥]𝜑 → (𝑥 = 𝑡 → ∃𝑦𝜑)) |
10 | ax5e 1909 | . 2 ⊢ (∃𝑦𝜑 → 𝜑) | |
11 | 9, 10 | syl6com 37 | 1 ⊢ (𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1531 ∃wex 1776 [wsb 2065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-12 2173 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-sb 2066 |
This theorem is referenced by: stdpc7 2248 sbequ12 2249 sb1OLD 2503 sb4a 2505 dfsb1 2506 dfsb2 2528 sbequiOLD 2530 sbi1OLD 2538 bj-ssbid2 33990 2pm13.193 40879 2pm13.193VD 41230 |
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