![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 2albii | Structured version Visualization version GIF version |
Description: Inference adding two universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.) |
Ref | Expression |
---|---|
albii.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
2albii | ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑥∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | albii.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
2 | 1 | albii 1815 | . 2 ⊢ (∀𝑦𝜑 ↔ ∀𝑦𝜓) |
3 | 2 | albii 1815 | 1 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑥∀𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∀wal 1534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 |
This theorem depends on definitions: df-bi 207 |
This theorem is referenced by: 3albii 1817 sbcom2 2170 2sb6rf 2475 mo4f 2564 2mo2 2644 2mos 2646 2mosOLD 2647 r3al 3194 ralcom 3286 ralcomf 3299 sbccomlem 3877 nfnid 5380 ssrel3 5798 raliunxp 5852 cnvsym 6134 cnvsymOLD 6135 cnvsymOLDOLD 6136 intasym 6137 intirr 6140 codir 6142 qfto 6143 dfpo2 6317 dffun4 6578 fun11 6641 fununi 6642 mpo2eqb 7564 frpoins3xpg 8163 xpord3inddlem 8177 aceq0 10155 zfac 10497 zfcndac 10656 addsrmo 11110 mulsrmo 11111 cotr2g 15011 isirred2 20437 isdomn3 20731 bnj580 34905 bnj978 34941 axacprim 35686 dfso2 35734 dfon2lem8 35771 dffun10 35895 wl-sbcom2d 37541 mpobi123f 38148 r2alan 38230 inxpss 38292 inxpss3 38295 cnvref5 38332 trcoss2 38465 dfantisymrel5 38743 antisymrelres 38744 dford4 43017 undmrnresiss 43593 cnvssco 43595 pm14.12 44416 ichn 47380 dfich2 47382 ichcom 47383 ichbi12i 47384 |
Copyright terms: Public domain | W3C validator |