Mathbox for Wolf Lammen |
< Previous
Next >
Nearby theorems |
||
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 261 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜑)) | |
2 | 1 | dral1 2434 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜑)) |
3 | 2 | aecoms 2423 | . . 3 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 ↔ ∀𝑦𝜑)) |
4 | 3 | dral1 2434 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑)) |
5 | 4 | aecoms 2423 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-10 2132 ax-12 2166 ax-13 2367 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-ex 1778 df-nf 1782 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |