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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sbex | Structured version Visualization version GIF version |
Description: If a proposition is true for a specific instance, then there exists an instance such that it is true for it. See spsbe 1871. (Contributed by BJ, 22-Dec-2020.) |
Ref | Expression |
---|---|
bj-sbex | ⊢ ([𝑡/𝑥]b𝜑 → ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ssb 31809 | . . 3 ⊢ ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
2 | ax6ev 1877 | . . . 4 ⊢ ∃𝑦 𝑦 = 𝑡 | |
3 | exim 1751 | . . . 4 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → (∃𝑦 𝑦 = 𝑡 → ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
4 | 2, 3 | mpi 20 | . . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑)) |
5 | 1, 4 | sylbi 206 | . 2 ⊢ ([𝑡/𝑥]b𝜑 → ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑)) |
6 | exim 1751 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑)) | |
7 | 6 | eximi 1752 | . 2 ⊢ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑦(∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑)) |
8 | ax6ev 1877 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝑦 | |
9 | pm2.27 41 | . . . 4 ⊢ (∃𝑥 𝑥 = 𝑦 → ((∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑) → ∃𝑥𝜑)) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ ((∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑) → ∃𝑥𝜑) |
11 | 10 | exlimiv 1845 | . 2 ⊢ (∃𝑦(∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑) → ∃𝑥𝜑) |
12 | 5, 7, 11 | 3syl 18 | 1 ⊢ ([𝑡/𝑥]b𝜑 → ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 ∃wex 1695 [wssb 31808 |
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 |
This theorem depends on definitions: df-bi 196 df-ex 1696 df-ssb 31809 |
This theorem is referenced by: bj-ssbft 31831 |
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