![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 2thd | Structured version Visualization version GIF version |
Description: Two truths are equivalent. Deduction form. (Contributed by NM, 3-Jun-2012.) |
Ref | Expression |
---|---|
2thd.1 | ⊢ (𝜑 → 𝜓) |
2thd.2 | ⊢ (𝜑 → 𝜒) |
Ref | Expression |
---|---|
2thd | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2thd.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | 2thd.2 | . 2 ⊢ (𝜑 → 𝜒) | |
3 | pm5.1im 263 | . 2 ⊢ (𝜓 → (𝜒 → (𝜓 ↔ 𝜒))) | |
4 | 1, 2, 3 | sylc 65 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 |
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 207 |
This theorem is referenced by: 2falsed 376 biort 935 vtocl2d 3561 vtocl2 3565 vtocl3 3566 rspcime 3626 sbc2or 3799 disjprg 5143 euotd 5522 posn 5773 frsn 5775 cnvpo 6308 elabrex 7261 elabrexg 7262 riota5f 7415 smoord 8403 brwdom2 9610 finacn 10087 acacni 10178 dfac13 10180 fin1a2lem10 10446 gch2 10712 gchac 10718 recmulnq 11001 nn1m1nn 12284 nn0sub 12573 xnn0n0n1ge2b 13170 qextltlem 13240 xnn0lem1lt 13282 xsubge0 13299 xlesubadd 13301 iccshftr 13522 iccshftl 13524 iccdil 13526 icccntr 13528 fzaddel 13594 elfzomelpfzo 13806 sqlecan 14244 nnesq 14262 hashdom 14414 swrdspsleq 14699 repswsymballbi 14814 m1exp1 16409 bitsmod 16469 dvdssq 16600 pcdvdsb 16902 vdwmc2 17012 acsfn 17703 subsubc 17903 funcres2b 17947 isipodrs 18594 issubg3 19174 sdrgacs 20818 lmhmlvec 21126 opnnei 23143 lmss 23321 lmres 23323 cmpfi 23431 xkopt 23678 acufl 23940 lmhmclm 25133 equivcmet 25364 degltlem1 26125 mdegle0 26130 cxple2 26753 rlimcnp3 27024 dchrelbas3 27296 tgcolg 28576 hlbtwn 28633 eupth2lem3lem6 30261 ifnebib 32569 isoun 32716 unitprodclb 33396 smatrcl 33756 msrrcl 35527 fz0n 35710 onint1 36431 bj-animbi 36541 bj-nfcsym 36881 matunitlindf 37604 ftc1anclem6 37684 lcvexchlem1 39015 ltrnatb 40119 cdlemg27b 40678 dvdsexpnn0 42347 fsuppind 42576 gicabl 43087 dfacbasgrp 43096 rp-fakeimass 43501 or3or 44012 radcnvrat 44309 eliooshift 45458 ellimcabssub0 45572 |
Copyright terms: Public domain | W3C validator |