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| Mirrors > Home > MPE Home > Th. List > rexlimivOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of rexlimiv 3147 as of 19-Dec-2024.) (Contributed by NM, 20-Nov-1994.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| rexlimivOLD.1 | ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) | 
| Ref | Expression | 
|---|---|
| rexlimivOLD | ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rexlimivOLD.1 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) | |
| 2 | 1 | rgen 3062 | . 2 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) | 
| 3 | r19.23v 3182 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)) | |
| 4 | 2, 3 | mpbi 230 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-ral 3061 df-rex 3070 | 
| This theorem is referenced by: (None) | 
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