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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sbfvv | Structured version Visualization version GIF version |
Description: Version of sbf 2368 with two dv conditions, which does not require ax-10 2006 nor ax-13 2234. (Contributed by BJ, 1-May-2021.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-sbfvv | ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spsbe 1871 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑) | |
2 | 19.9v 1883 | . . 3 ⊢ (∃𝑥𝜑 ↔ 𝜑) | |
3 | 1, 2 | sylib 207 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → 𝜑) |
4 | ax-5 1827 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
5 | bj-stdpc4v 31942 | . . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → [𝑦 / 𝑥]𝜑) |
7 | 3, 6 | impbii 198 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∀wal 1473 ∃wex 1695 [wsb 1867 |
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-an 385 df-ex 1696 df-sb 1868 |
This theorem is referenced by: bj-vjust2 32206 |
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