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Mirrors > Home > MPE Home > Th. List > cbvaev | Structured version Visualization version GIF version |
Description: Change bound variable in an equality with a dv condition. Instance of aev 1970. (Contributed by NM, 22-Jul-2015.) (Revised by BJ, 18-Jun-2019.) |
Ref | Expression |
---|---|
cbvaev | ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax7 1930 | . . 3 ⊢ (𝑥 = 𝑡 → (𝑥 = 𝑦 → 𝑡 = 𝑦)) | |
2 | 1 | cbvalivw 1921 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑡 𝑡 = 𝑦) |
3 | ax7 1930 | . . 3 ⊢ (𝑡 = 𝑧 → (𝑡 = 𝑦 → 𝑧 = 𝑦)) | |
4 | 3 | cbvalivw 1921 | . 2 ⊢ (∀𝑡 𝑡 = 𝑦 → ∀𝑧 𝑧 = 𝑦) |
5 | 2, 4 | syl 17 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: aevlem0 1967 aevlem 1968 axc11nlemOLD2 1975 aevlemOLD 1976 axc11nlemOLD 2146 axc11nlemALT 2294 aevlemALTOLD 2308 |
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