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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-hbaeb | Structured version Visualization version GIF version |
Description: Biconditional version of hbae 2434. (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-hbaeb | ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧∀𝑥 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-hbaeb2 36801 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧 𝑥 = 𝑦) | |
2 | alcom 2157 | . 2 ⊢ (∀𝑥∀𝑧 𝑥 = 𝑦 ↔ ∀𝑧∀𝑥 𝑥 = 𝑦) | |
3 | 1, 2 | bitri 275 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧∀𝑥 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∀wal 1535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-10 2139 ax-11 2155 ax-12 2175 ax-13 2375 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 |
This theorem is referenced by: (None) |
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