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Mirrors > Home > MPE Home > Th. List > pm5.21nd | Structured version Visualization version GIF version |
Description: Eliminate an antecedent implied by each side of a biconditional. Variant of pm5.21ndd 384. (Contributed by NM, 20-Nov-2005.) (Proof shortened by Wolf Lammen, 4-Nov-2013.) |
Ref | Expression |
---|---|
pm5.21nd.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝜃) |
pm5.21nd.2 | ⊢ ((𝜑 ∧ 𝜒) → 𝜃) |
pm5.21nd.3 | ⊢ (𝜃 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
pm5.21nd | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.21nd.1 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜃) | |
2 | 1 | ex 416 | . 2 ⊢ (𝜑 → (𝜓 → 𝜃)) |
3 | pm5.21nd.2 | . . 3 ⊢ ((𝜑 ∧ 𝜒) → 𝜃) | |
4 | 3 | ex 416 | . 2 ⊢ (𝜑 → (𝜒 → 𝜃)) |
5 | pm5.21nd.3 | . . 3 ⊢ (𝜃 → (𝜓 ↔ 𝜒)) | |
6 | 5 | a1i 11 | . 2 ⊢ (𝜑 → (𝜃 → (𝜓 ↔ 𝜒))) |
7 | 2, 4, 6 | pm5.21ndd 384 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 210 df-an 400 |
This theorem is referenced by: ideqg 5720 fvelimab 6784 brrpssg 7513 ordsucelsuc 7601 releldm2 7814 relbrtpos 7979 relelec 8436 elfiun 9046 fpwwe2lem2 10246 fpwwelem 10259 fzrev3 13178 elfzp12 13191 eqgval 18593 eltg 21854 eltg2 21855 cncnp2 22178 isref 22406 islocfin 22414 opeldifid 30657 isfne 34265 copsex2b 35046 bj-ideqgALT 35064 bj-idreseq 35068 bj-ideqg1ALT 35071 opelopab3 35612 isdivrngo 35845 brssr 36356 islshpkrN 36871 dihatexv2 39090 |
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