|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > ralrexbidOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of ralrexbid 3105 as of 4-Nov-2024. (Contributed by AV, 21-Oct-2023.) Reduce axiom usage. (Revised by SN, 13-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| ralrexbid.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜃)) | 
| Ref | Expression | 
|---|---|
| ralrexbidOLD | ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜃)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-ral 3061 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
| 2 | ralrexbid.1 | . . . . . 6 ⊢ (𝜑 → (𝜓 ↔ 𝜃)) | |
| 3 | 2 | imim2i 16 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜃))) | 
| 4 | 3 | pm5.32d 577 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜃))) | 
| 5 | 4 | alexbii 1832 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜃))) | 
| 6 | 1, 5 | sylbi 217 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜃))) | 
| 7 | df-rex 3070 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
| 8 | df-rex 3070 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜃 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜃)) | |
| 9 | 6, 7, 8 | 3bitr4g 314 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜃)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1537 ∃wex 1778 ∈ 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 | 
| 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) | 
| Copyright terms: Public domain | W3C validator |