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 1799 | . 2 ⊢ ∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) |
4 | sp 2176 | . . . . 5 ⊢ (∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
5 | 4 | alcoms 2155 | . . . 4 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
6 | wl-equsal1t 35686 | . . . 4 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)) | |
7 | 5, 6 | syl5ib 243 | . . 3 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) → 𝜑)) |
8 | wl-equsalcom 35687 | . . . . 5 ⊢ (∀𝑦(𝑦 = 𝑥 → 𝜑) ↔ ∀𝑦(𝑥 = 𝑦 → 𝜑)) | |
9 | wl-equsal1t 35686 | . . . . . 6 ⊢ (Ⅎ𝑦𝜑 → (∀𝑦(𝑦 = 𝑥 → 𝜑) ↔ 𝜑)) | |
10 | 9 | biimpd 228 | . . . . 5 ⊢ (Ⅎ𝑦𝜑 → (∀𝑦(𝑦 = 𝑥 → 𝜑) → 𝜑)) |
11 | 8, 10 | syl5bir 242 | . . . 4 ⊢ (Ⅎ𝑦𝜑 → (∀𝑦(𝑥 = 𝑦 → 𝜑) → 𝜑)) |
12 | 11 | spsd 2180 | . . 3 ⊢ (Ⅎ𝑦𝜑 → (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) → 𝜑)) |
13 | 7, 12 | jaoi 854 | . 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 844 ∀wal 1537 Ⅎwnf 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-11 2154 ax-12 2171 ax-13 2372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ex 1783 df-nf 1787 |
This theorem is referenced by: (None) |
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