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Mirrors > Home > MPE Home > Th. List > spsbe | Structured version Visualization version GIF version |
Description: Existential generalization: if a proposition is true for a specific instance, then there exists an instance where it is true. (Contributed by NM, 29-Jun-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.) Revise df-sb 2016. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 11-Jul-2023.) |
Ref | Expression |
---|---|
spsbe | ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sb 2016 | . . 3 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
2 | ax6ev 1930 | . . . 4 ⊢ ∃𝑦 𝑦 = 𝑡 | |
3 | exim 1796 | . . . 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 209 | . 2 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑)) |
6 | exsbim 1958 | . 2 ⊢ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑) | |
7 | 5, 6 | syl 17 | 1 ⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1505 ∃wex 1742 [wsb 2015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 |
This theorem depends on definitions: df-bi 199 df-ex 1743 df-sb 2016 |
This theorem is referenced by: sbv 2040 sbft 2198 2mo 2680 noel 4184 bj-ssbft 33500 wl-lem-moexsb 34242 nsb 38542 spsbce-2 40126 sb5ALT 40275 sb5ALTVD 40663 |
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