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Mirrors > Home > NFE Home > Th. List > r19.40 | GIF version |
Description: Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.) |
Ref | Expression |
---|---|
r19.40 | ⊢ (∃x ∈ A (φ ∧ ψ) → (∃x ∈ A φ ∧ ∃x ∈ A ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 443 | . . 3 ⊢ ((φ ∧ ψ) → φ) | |
2 | 1 | reximi 2722 | . 2 ⊢ (∃x ∈ A (φ ∧ ψ) → ∃x ∈ A φ) |
3 | simpr 447 | . . 3 ⊢ ((φ ∧ ψ) → ψ) | |
4 | 3 | reximi 2722 | . 2 ⊢ (∃x ∈ A (φ ∧ ψ) → ∃x ∈ A ψ) |
5 | 2, 4 | jca 518 | 1 ⊢ (∃x ∈ A (φ ∧ ψ) → (∃x ∈ A φ ∧ ∃x ∈ A ψ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∃wrex 2616 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ex 1542 df-ral 2620 df-rex 2621 |
This theorem is referenced by: (None) |
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