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Mirrors > Home > MPE Home > Th. List > equs4v | Structured version Visualization version GIF version |
Description: Version of equs4 2434 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 1968 | . 2 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | exintr 1889 | . 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 398 ∀wal 1531 ∃wex 1776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-6 1966 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 |
This theorem is referenced by: sb1v 2091 sb2vOLDOLD 2508 sb2vOLDALT 2579 bj-sb56 33989 bj-equs45fv 34128 |
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