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Mirrors > Home > MPE Home > Th. List > Mathboxes > sb5ALT | Structured version Visualization version GIF version |
Description: Equivalence for substitution. Alternate proof of sb5 2251. This proof is sb5ALTVD 40082 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sb5ALT | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equsb1 2444 | . . . 4 ⊢ [𝑦 / 𝑥]𝑥 = 𝑦 | |
2 | sban 2475 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝑥 = 𝑦 ∧ 𝜑) ↔ ([𝑦 / 𝑥]𝑥 = 𝑦 ∧ [𝑦 / 𝑥]𝜑)) | |
3 | 2 | simplbi2com 498 | . . . 4 ⊢ ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝑥 = 𝑦 → [𝑦 / 𝑥](𝑥 = 𝑦 ∧ 𝜑))) |
4 | 1, 3 | mpi 20 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝑥 = 𝑦 ∧ 𝜑)) |
5 | spsbe 2015 | . . 3 ⊢ ([𝑦 / 𝑥](𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
7 | hbs1 2255 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | |
8 | simpr 479 | . . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝜑) | |
9 | 8 | a1i 11 | . . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ((𝑥 = 𝑦 ∧ 𝜑) → 𝜑)) |
10 | simpl 476 | . . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝑥 = 𝑦) | |
11 | 10 | a1i 11 | . . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ((𝑥 = 𝑦 ∧ 𝜑) → 𝑥 = 𝑦)) |
12 | sbequ1 2228 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑)) | |
13 | 12 | com12 32 | . . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → [𝑦 / 𝑥]𝜑)) |
14 | 9, 11, 13 | syl6c 70 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ((𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑)) |
15 | 7, 14 | exlimexi 39684 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑) |
16 | 6, 15 | impbii 201 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∃wex 1823 [wsb 2011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-10 2135 ax-12 2163 ax-13 2334 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-ex 1824 df-nf 1828 df-sb 2012 |
This theorem is referenced by: (None) |
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