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Mirrors > Home > ILE Home > Th. List > pm5.21ndd | GIF version |
Description: Eliminate an antecedent implied by each side of a biconditional, deduction version. (Contributed by Paul Chapman, 21-Nov-2012.) (Revised by Mario Carneiro, 31-Jan-2015.) |
Ref | Expression |
---|---|
pm5.21ndd.1 | ⊢ (𝜑 → (𝜒 → 𝜓)) |
pm5.21ndd.2 | ⊢ (𝜑 → (𝜃 → 𝜓)) |
pm5.21ndd.3 | ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃))) |
Ref | Expression |
---|---|
pm5.21ndd | ⊢ (𝜑 → (𝜒 ↔ 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.21ndd.1 | . . . 4 ⊢ (𝜑 → (𝜒 → 𝜓)) | |
2 | pm5.21ndd.3 | . . . 4 ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃))) | |
3 | 1, 2 | syld 44 | . . 3 ⊢ (𝜑 → (𝜒 → (𝜒 ↔ 𝜃))) |
4 | 3 | ibd 176 | . 2 ⊢ (𝜑 → (𝜒 → 𝜃)) |
5 | pm5.21ndd.2 | . . . . 5 ⊢ (𝜑 → (𝜃 → 𝜓)) | |
6 | 5, 2 | syld 44 | . . . 4 ⊢ (𝜑 → (𝜃 → (𝜒 ↔ 𝜃))) |
7 | bicom1 129 | . . . 4 ⊢ ((𝜒 ↔ 𝜃) → (𝜃 ↔ 𝜒)) | |
8 | 6, 7 | syl6 33 | . . 3 ⊢ (𝜑 → (𝜃 → (𝜃 ↔ 𝜒))) |
9 | 8 | ibd 176 | . 2 ⊢ (𝜑 → (𝜃 → 𝜒)) |
10 | 4, 9 | impbid 127 | 1 ⊢ (𝜑 → (𝜒 ↔ 𝜃)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: pm5.21nd 858 sbcrext 2891 rmob 2906 epelg 4045 eqbrrdva 4523 relbrcnvg 4724 fmptco 5351 ovelrn 5669 brtpos2 5889 brdomg 6252 genpelvl 6702 genpelvu 6703 fzoval 9158 clim 10120 |
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