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| Description: Convert a restricted existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.) | 
| Ref | Expression | 
|---|---|
| ralpr.1 | ⊢ 𝐴 ∈ V | 
| ralpr.2 | ⊢ 𝐵 ∈ V | 
| ralpr.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | 
| ralpr.4 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) | 
| Ref | Expression | 
|---|---|
| rexpr | ⊢ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ralpr.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | ralpr.2 | . 2 ⊢ 𝐵 ∈ V | |
| 3 | ralpr.3 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 4 | ralpr.4 | . . 3 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) | |
| 5 | 3, 4 | rexprg 4696 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒))) | 
| 6 | 1, 2, 5 | mp2an 692 | 1 ⊢ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∨ wo 847 = wceq 1539 ∈ wcel 2107 ∃wrex 3069 Vcvv 3479 {cpr 4627 | 
| 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 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-v 3481 df-un 3955 df-sn 4626 df-pr 4628 | 
| This theorem is referenced by: xpsdsval 24392 poimir 37661 | 
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