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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 2275. This proof is sb5ALTVD 42055 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 2495 | . . . 4 ⊢ [𝑦 / 𝑥]𝑥 = 𝑦 | |
2 | sban 2089 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝑥 = 𝑦 ∧ 𝜑) ↔ ([𝑦 / 𝑥]𝑥 = 𝑦 ∧ [𝑦 / 𝑥]𝜑)) | |
3 | 2 | simplbi2com 506 | . . . 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 2091 | . . 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 2273 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | |
8 | simpr 488 | . . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝜑) | |
9 | 8 | a1i 11 | . . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ((𝑥 = 𝑦 ∧ 𝜑) → 𝜑)) |
10 | simpl 486 | . . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝑥 = 𝑦) | |
11 | 10 | a1i 11 | . . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ((𝑥 = 𝑦 ∧ 𝜑) → 𝑥 = 𝑦)) |
12 | sbequ1 2248 | . . . . 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 41666 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑) |
16 | 6, 15 | impbii 212 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∃wex 1786 [wsb 2073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-10 2144 ax-12 2178 ax-13 2371 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ex 1787 df-nf 1791 df-sb 2074 |
This theorem is referenced by: (None) |
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