Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-ax11-lem9 | Structured version Visualization version GIF version |
Description: The easy part when 𝑥 coincides with 𝑦. (Contributed by Wolf Lammen, 30-Jun-2019.) |
Ref | Expression |
---|---|
wl-ax11-lem9 | ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biidd 265 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜑)) | |
2 | 1 | dral1 2439 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜑)) |
3 | 2 | aecoms 2428 | . . 3 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 ↔ ∀𝑦𝜑)) |
4 | 3 | dral1 2439 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑)) |
5 | 4 | aecoms 2428 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1540 |
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 1917 ax-6 1975 ax-7 2020 ax-10 2145 ax-12 2179 ax-13 2372 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ex 1787 df-nf 1791 |
This theorem is referenced by: (None) |
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