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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 373 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) |
3 | 2 | rexlimdva 2581 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2135 ∃wrex 2443 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1434 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-4 1497 ax-17 1513 ax-ial 1521 ax-i5r 1522 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-ral 2447 df-rex 2448 |
This theorem is referenced by: rexlimddv 2586 nnsucuniel 6454 omp1eomlem 7050 ctmlemr 7064 mulgt0sr 7710 axpre-suploclemres 7833 cnegex 8067 receuap 8557 rexanuz 10916 climcaucn 11278 fsumiun 11404 dvdsval2 11716 prmind2 12031 pcprmpw2 12243 pockthg 12266 tgcl 12611 neiint 12692 restopnb 12728 iscnp4 12765 blssexps 12976 blssex 12977 |
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