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Mirrors > Home > ILE Home > Th. List > orbi1d | GIF version |
Description: Deduction adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
orbid.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
orbi1d | ⊢ (𝜑 → ((𝜓 ∨ 𝜃) ↔ (𝜒 ∨ 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orbid.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | orbi2d 780 | . 2 ⊢ (𝜑 → ((𝜃 ∨ 𝜓) ↔ (𝜃 ∨ 𝜒))) |
3 | orcom 718 | . 2 ⊢ ((𝜓 ∨ 𝜃) ↔ (𝜃 ∨ 𝜓)) | |
4 | orcom 718 | . 2 ⊢ ((𝜒 ∨ 𝜃) ↔ (𝜃 ∨ 𝜒)) | |
5 | 2, 3, 4 | 3bitr4g 222 | 1 ⊢ (𝜑 → ((𝜓 ∨ 𝜃) ↔ (𝜒 ∨ 𝜃))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∨ wo 698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: orbi1 782 orbi12d 783 xorbi1d 1360 eueq2dc 2861 uneq1 3228 r19.45mv 3461 rexprg 3583 rextpg 3585 swopolem 4235 sowlin 4250 onsucelsucexmidlem1 4451 onsucelsucexmid 4453 ordsoexmid 4485 isosolem 5733 acexmidlema 5773 acexmidlemb 5774 acexmidlem2 5779 acexmidlemv 5780 freceq1 6297 exmidaclem 7081 exmidac 7082 elinp 7306 prloc 7323 suplocexprlemloc 7553 ltsosr 7596 suplocsrlemb 7638 axpre-ltwlin 7715 axpre-suploclemres 7733 axpre-suploc 7734 apreap 8373 apreim 8389 sup3exmid 8739 nn01to3 9436 ltxr 9592 fzpr 9888 elfzp12 9910 lcmval 11780 lcmass 11802 isprm6 11861 dedekindeulemloc 12805 dedekindeulemeu 12808 suplociccreex 12810 dedekindicclemloc 12814 dedekindicclemeu 12817 |
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