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Mirrors > Home > NFE Home > Th. List > r19.29r | GIF version |
Description: Variation of Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.) |
Ref | Expression |
---|---|
r19.29r | ⊢ ((∃x ∈ A φ ∧ ∀x ∈ A ψ) → ∃x ∈ A (φ ∧ ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.29 2754 | . 2 ⊢ ((∀x ∈ A ψ ∧ ∃x ∈ A φ) → ∃x ∈ A (ψ ∧ φ)) | |
2 | ancom 437 | . 2 ⊢ ((∃x ∈ A φ ∧ ∀x ∈ A ψ) ↔ (∀x ∈ A ψ ∧ ∃x ∈ A φ)) | |
3 | ancom 437 | . . 3 ⊢ ((φ ∧ ψ) ↔ (ψ ∧ φ)) | |
4 | 3 | rexbii 2639 | . 2 ⊢ (∃x ∈ A (φ ∧ ψ) ↔ ∃x ∈ A (ψ ∧ φ)) |
5 | 1, 2, 4 | 3imtr4i 257 | 1 ⊢ ((∃x ∈ A φ ∧ ∀x ∈ A ψ) → ∃x ∈ A (φ ∧ ψ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∀wral 2614 ∃wrex 2615 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-11 1746 |
This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-ral 2619 df-rex 2620 |
This theorem is referenced by: 2reu5 3044 |
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