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Mirrors > Home > NFE Home > Th. List > reximi | GIF version |
Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 18-Oct-1996.) |
Ref | Expression |
---|---|
reximi.1 | ⊢ (φ → ψ) |
Ref | Expression |
---|---|
reximi | ⊢ (∃x ∈ A φ → ∃x ∈ A ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reximi.1 | . . 3 ⊢ (φ → ψ) | |
2 | 1 | a1i 10 | . 2 ⊢ (x ∈ A → (φ → ψ)) |
3 | 2 | reximia 2719 | 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-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 2619 df-rex 2620 |
This theorem is referenced by: r19.40 2762 reu3 3026 2reu5 3044 ssiun 4008 iinss 4017 lefinlteq 4463 sucevenodd 4510 sfinltfin 4535 vfinspsslem1 4550 pw1fin 6169 addlec 6208 nncdiv3 6277 |
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