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Mirrors > Home > ILE Home > Th. List > rexlimdvaa | GIF version |
Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Mario Carneiro, 15-Jun-2016.) |
Ref | Expression |
---|---|
rexlimdvaa.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝜒) |
Ref | Expression |
---|---|
rexlimdvaa | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexlimdvaa.1 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝜒) | |
2 | 1 | expr 357 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) |
3 | 2 | rexlimdva 2433 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1393 ∃wrex 2307 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-17 1419 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-ral 2311 df-rex 2312 |
This theorem is referenced by: rexlimddv 2437 mulgt0sr 6862 cnegex 7189 receuap 7650 rexanuz 9587 climcaucn 9870 |
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