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Mirrors > Home > MPE Home > Th. List > rexprg | Structured version Visualization version GIF version |
Description: Convert a restricted existential quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) (Proof shortened by AV, 8-Apr-2023.) |
Ref | Expression |
---|---|
ralprg.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
ralprg.2 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) |
Ref | Expression |
---|---|
rexprg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1911 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | nfv 1911 | . 2 ⊢ Ⅎ𝑥𝜒 | |
3 | ralprg.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | ralprg.2 | . 2 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) | |
5 | 1, 2, 3, 4 | rexprgf 4624 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 {cpr 4562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-v 3496 df-sbc 3772 df-un 3940 df-sn 4561 df-pr 4563 |
This theorem is referenced by: rextpg 4628 rexpr 4630 reurexprg 4633 fr2nr 5527 sgrp2nmndlem5 18088 nb3grprlem2 27157 nfrgr2v 28045 3vfriswmgrlem 28050 brfvrcld 40029 rnmptpr 41426 ldepspr 44522 zlmodzxzldeplem4 44552 |
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