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Mirrors > Home > NFE Home > Th. List > reximia | GIF version |
Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.) |
Ref | Expression |
---|---|
reximia.1 | ⊢ (x ∈ A → (φ → ψ)) |
Ref | Expression |
---|---|
reximia | ⊢ (∃x ∈ A φ → ∃x ∈ A ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexim 2718 | . 2 ⊢ (∀x ∈ A (φ → ψ) → (∃x ∈ A φ → ∃x ∈ A ψ)) | |
2 | reximia.1 | . 2 ⊢ (x ∈ A → (φ → ψ)) | |
3 | 1, 2 | mprg 2683 | 1 ⊢ (∃x ∈ A φ → ∃x ∈ A ψ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1710 ∃wrex 2615 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ex 1542 df-ral 2619 df-rex 2620 |
This theorem is referenced by: reximi 2721 |
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