| Mathbox for Wolf Lammen |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-equsal1i | Structured version Visualization version GIF version | ||
| Description: The antecedent 𝑥 = 𝑦 is irrelevant, if one or both setvar variables are not free in 𝜑. (Contributed by Wolf Lammen, 1-Sep-2018.) |
| Ref | Expression |
|---|---|
| wl-equsal1i.1 | ⊢ (Ⅎ𝑥𝜑 ∨ Ⅎ𝑦𝜑) |
| wl-equsal1i.2 | ⊢ (𝑥 = 𝑦 → 𝜑) |
| Ref | Expression |
|---|---|
| wl-equsal1i | ⊢ 𝜑 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | wl-equsal1i.1 | . 2 ⊢ (Ⅎ𝑥𝜑 ∨ Ⅎ𝑦𝜑) | |
| 2 | wl-equsal1i.2 | . . 3 ⊢ (𝑥 = 𝑦 → 𝜑) | |
| 3 | 2 | gen2 1796 | . 2 ⊢ ∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) |
| 4 | sp 2183 | . . . . 5 ⊢ (∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
| 5 | 4 | alcoms 2158 | . . . 4 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
| 6 | wl-equsal1t 37543 | . . . 4 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)) | |
| 7 | 5, 6 | imbitrid 244 | . . 3 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) → 𝜑)) |
| 8 | wl-equsalcom 37544 | . . . . 5 ⊢ (∀𝑦(𝑦 = 𝑥 → 𝜑) ↔ ∀𝑦(𝑥 = 𝑦 → 𝜑)) | |
| 9 | wl-equsal1t 37543 | . . . . . 6 ⊢ (Ⅎ𝑦𝜑 → (∀𝑦(𝑦 = 𝑥 → 𝜑) ↔ 𝜑)) | |
| 10 | 9 | biimpd 229 | . . . . 5 ⊢ (Ⅎ𝑦𝜑 → (∀𝑦(𝑦 = 𝑥 → 𝜑) → 𝜑)) |
| 11 | 8, 10 | biimtrrid 243 | . . . 4 ⊢ (Ⅎ𝑦𝜑 → (∀𝑦(𝑥 = 𝑦 → 𝜑) → 𝜑)) |
| 12 | 11 | spsd 2187 | . . 3 ⊢ (Ⅎ𝑦𝜑 → (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) → 𝜑)) |
| 13 | 7, 12 | jaoi 858 | . 2 ⊢ ((Ⅎ𝑥𝜑 ∨ Ⅎ𝑦𝜑) → (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) → 𝜑)) |
| 14 | 1, 3, 13 | mp2 9 | 1 ⊢ 𝜑 |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∨ wo 848 ∀wal 1538 Ⅎwnf 1783 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: (None) |
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