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Mirrors > Home > ILE Home > Th. List > elv | GIF version |
Description: Technical lemma used to shorten proofs. If a proposition is implied by 𝑥 ∈ V (which is true, see vex 2741), then it is true. (Contributed by Peter Mazsa, 13-Oct-2018.) |
Ref | Expression |
---|---|
elv.1 | ⊢ (𝑥 ∈ V → 𝜑) |
Ref | Expression |
---|---|
elv | ⊢ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2741 | . 2 ⊢ 𝑥 ∈ V | |
2 | elv.1 | . 2 ⊢ (𝑥 ∈ V → 𝜑) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝜑 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 Vcvv 2738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-v 2740 |
This theorem is referenced by: xpiindim 4765 disjxp1 6237 ixpiinm 6724 ixpsnf1o 6736 iunfidisj 6945 ssfii 6973 fifo 6979 dcfi 6980 omp1eomlem 7093 exmidomniim 7139 bcval5 10743 rexfiuz 10998 fsum2dlemstep 11442 fsumcnv 11445 fisumcom2 11446 fsumconst 11462 modfsummodlemstep 11465 fsumabs 11473 fprodcllemf 11621 fprod2dlemstep 11630 fprodcnv 11633 fprodcom2fi 11634 fprodmodd 11649 ennnfonelemim 12425 topnfn 12693 ptex 12713 xpsff1o 12768 ismgm 12776 issgrp 12809 ismnddef 12819 isnsg 13062 fnmgp 13132 isring 13183 iuncld 13618 txbas 13761 txdis 13780 xmetunirn 13861 xmettxlem 14012 xmettx 14013 pw1nct 14755 |
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