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| Mirrors > Home > MPE Home > Th. List > equs4v | Structured version Visualization version GIF version | ||
| Description: Version of equs4 2421 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 10-May-1993.) (Revised by BJ, 31-May-2019.) |
| Ref | Expression |
|---|---|
| equs4v | ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax6ev 1969 | . 2 ⊢ ∃𝑥 𝑥 = 𝑦 | |
| 2 | exintr 1892 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
| 3 | 1, 2 | mpi 20 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 ∃wex 1779 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-6 1967 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 |
| This theorem is referenced by: sb1v 2087 sbalex 2242 sbalexOLD 2243 equsexv 2268 bj-subst 36662 bj-equs45fv 36812 |
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