Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > 19.29r | GIF version |
Description: Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.) |
Ref | Expression |
---|---|
19.29r | ⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.29 1607 | . 2 ⊢ ((∀𝑥𝜓 ∧ ∃𝑥𝜑) → ∃𝑥(𝜓 ∧ 𝜑)) | |
2 | ancom 264 | . 2 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (∀𝑥𝜓 ∧ ∃𝑥𝜑)) | |
3 | exancom 1595 | . 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑)) | |
4 | 1, 2, 3 | 3imtr4i 200 | 1 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1340 ∃wex 1479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1434 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-4 1497 ax-ial 1521 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: 19.29r2 1609 19.29x 1610 exan 1680 ax9o 1685 equvini 1745 eu2 2057 intab 3847 imadiflem 5261 bj-inex 13624 |
Copyright terms: Public domain | W3C validator |