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Mirrors > Home > NFE Home > Th. List > reximdv2 | GIF version |
Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 17-Sep-2003.) |
Ref | Expression |
---|---|
reximdv2.1 | ⊢ (φ → ((x ∈ A ∧ ψ) → (x ∈ B ∧ χ))) |
Ref | Expression |
---|---|
reximdv2 | ⊢ (φ → (∃x ∈ A ψ → ∃x ∈ B χ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reximdv2.1 | . . 3 ⊢ (φ → ((x ∈ A ∧ ψ) → (x ∈ B ∧ χ))) | |
2 | 1 | eximdv 1622 | . 2 ⊢ (φ → (∃x(x ∈ A ∧ ψ) → ∃x(x ∈ B ∧ χ))) |
3 | df-rex 2620 | . 2 ⊢ (∃x ∈ A ψ ↔ ∃x(x ∈ A ∧ ψ)) | |
4 | df-rex 2620 | . 2 ⊢ (∃x ∈ B χ ↔ ∃x(x ∈ B ∧ χ)) | |
5 | 2, 3, 4 | 3imtr4g 261 | 1 ⊢ (φ → (∃x ∈ A ψ → ∃x ∈ B χ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∃wex 1541 ∈ 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 |
This theorem depends on definitions: df-bi 177 df-ex 1542 df-rex 2620 |
This theorem is referenced by: ssrexv 3331 |
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