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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-chvarv | Structured version Visualization version GIF version |
Description: Version of chvar 2250 with a dv condition, which does not require ax-13 2234. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-chvarv.nf | ⊢ Ⅎ𝑥𝜓 |
bj-chvarv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
bj-chvarv.2 | ⊢ 𝜑 |
Ref | Expression |
---|---|
bj-chvarv | ⊢ 𝜓 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-chvarv.nf | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | bj-chvarv.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | biimpd 218 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
4 | 1, 3 | spimv1 2101 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) |
5 | bj-chvarv.2 | . 2 ⊢ 𝜑 | |
6 | 4, 5 | mpg 1715 | 1 ⊢ 𝜓 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 Ⅎwnf 1699 |
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 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-ex 1696 df-nf 1701 |
This theorem is referenced by: bj-axrep2 31977 bj-axrep3 31978 |
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