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Mirrors > Home > MPE Home > Th. List > mpgbir | Structured version Visualization version GIF version |
Description: Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.) |
Ref | Expression |
---|---|
mpgbir.1 | ⊢ (𝜑 ↔ ∀𝑥𝜓) |
mpgbir.2 | ⊢ 𝜓 |
Ref | Expression |
---|---|
mpgbir | ⊢ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpgbir.2 | . . 3 ⊢ 𝜓 | |
2 | 1 | ax-gen 1798 | . 2 ⊢ ∀𝑥𝜓 |
3 | mpgbir.1 | . 2 ⊢ (𝜑 ↔ ∀𝑥𝜓) | |
4 | 2, 3 | mpbir 230 | 1 ⊢ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wal 1540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 |
This theorem depends on definitions: df-bi 206 |
This theorem is referenced by: cvjust 2727 eqriv 2730 nfci 2887 abid2f 2937 abid2fOLD 2938 rgen 3064 rabeqcOLD 3682 ssriv 3987 ss2abiOLD 4065 nel0 4351 rab0 4383 abfOLD 4404 ssmin 4972 intab 4983 iunab 5055 iinab 5072 sndisj 5140 disjxsn 5142 intidOLD 5459 fr0 5656 relssi 5788 dmi 5922 dmep 5924 onfr 6404 funopabeq 6585 isarep2 6640 opabiotafun 6973 fvopab3ig 6995 opabex 7222 caovmo 7644 trom 7864 tz7.44lem1 8405 pwfir 9176 dfsup2 9439 zfregfr 9600 dfom3 9642 dfttrcl2 9719 trcl 9723 tc2 9737 rankf 9789 rankval4 9862 uniwun 10735 dfnn2 12225 dfuzi 12653 fzodisj 13666 fzodisjsn 13670 cycsubg 19085 efger 19586 made0 27368 lrrecfr 27427 ajfuni 30112 funadj 31139 rabexgfGS 31739 abrexdomjm 31744 ballotth 33536 bnj1133 34000 satfv0fun 34362 fmla0xp 34374 dfon3 34864 fnsingle 34891 dfiota3 34895 hftr 35154 bj-rabtrALT 35811 ismblfin 36529 abrexdom 36598 cllem0 42317 cotrintab 42365 brtrclfv2 42478 snhesn 42537 psshepw 42539 k0004val0 42905 scottabf 42999 compab 43201 onfrALT 43310 dvcosre 44628 upwordnul 45594 upwordsing 45598 cfsetssfset 45766 alimp-surprise 47827 |
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