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Mirrors > Home > NFE Home > Th. List > r19.41 | GIF version |
Description: Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 1-Nov-2010.) |
Ref | Expression |
---|---|
r19.41.1 | ⊢ Ⅎxψ |
Ref | Expression |
---|---|
r19.41 | ⊢ (∃x ∈ A (φ ∧ ψ) ↔ (∃x ∈ A φ ∧ ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anass 630 | . . . 4 ⊢ (((x ∈ A ∧ φ) ∧ ψ) ↔ (x ∈ A ∧ (φ ∧ ψ))) | |
2 | 1 | exbii 1582 | . . 3 ⊢ (∃x((x ∈ A ∧ φ) ∧ ψ) ↔ ∃x(x ∈ A ∧ (φ ∧ ψ))) |
3 | r19.41.1 | . . . 4 ⊢ Ⅎxψ | |
4 | 3 | 19.41 1879 | . . 3 ⊢ (∃x((x ∈ A ∧ φ) ∧ ψ) ↔ (∃x(x ∈ A ∧ φ) ∧ ψ)) |
5 | 2, 4 | bitr3i 242 | . 2 ⊢ (∃x(x ∈ A ∧ (φ ∧ ψ)) ↔ (∃x(x ∈ A ∧ φ) ∧ ψ)) |
6 | df-rex 2620 | . 2 ⊢ (∃x ∈ A (φ ∧ ψ) ↔ ∃x(x ∈ A ∧ (φ ∧ ψ))) | |
7 | df-rex 2620 | . . 3 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ)) | |
8 | 7 | anbi1i 676 | . 2 ⊢ ((∃x ∈ A φ ∧ ψ) ↔ (∃x(x ∈ A ∧ φ) ∧ ψ)) |
9 | 5, 6, 8 | 3bitr4i 268 | 1 ⊢ (∃x ∈ A (φ ∧ ψ) ↔ (∃x ∈ A φ ∧ ψ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∃wex 1541 Ⅎwnf 1544 ∈ wcel 1710 ∃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-6 1729 ax-11 1746 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ex 1542 df-nf 1545 df-rex 2620 |
This theorem is referenced by: r19.41v 2764 |
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