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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 379. (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 412 | . 2 ⊢ (𝜑 → (𝜓 → 𝜃)) |
3 | pm5.21nd.2 | . . 3 ⊢ ((𝜑 ∧ 𝜒) → 𝜃) | |
4 | 3 | ex 412 | . 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 379 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 |
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 206 df-an 396 |
This theorem is referenced by: ideqg 5854 fvelimab 6971 brrpssg 7730 ordsucelsuc 7825 releldm2 8047 relbrtpos 8243 relelec 8771 elfiun 9454 fpwwe2lem2 10656 fpwwelem 10669 fzrev3 13600 elfzp12 13613 eqgval 19132 eltg 22873 eltg2 22874 cncnp2 23198 isref 23426 islocfin 23434 opeldifid 32402 isfne 35823 copsex2b 36619 bj-ideqgALT 36637 bj-idreseq 36641 bj-ideqg1ALT 36644 opelopab3 37191 isdivrngo 37423 brssr 37973 islshpkrN 38592 dihatexv2 40812 |
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