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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 2692), 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 2692 | . 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 1481 Vcvv 2689 |
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-5 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-v 2691 |
This theorem is referenced by: xpiindim 4684 disjxp1 6141 ixpiinm 6626 ixpsnf1o 6638 iunfidisj 6842 ssfii 6870 fifo 6876 omp1eomlem 6987 exmidomniim 7021 bcval5 10541 rexfiuz 10793 fsum2dlemstep 11235 fsumcnv 11238 fisumcom2 11239 fsumconst 11255 modfsummodlemstep 11258 fsumabs 11266 ennnfonelemim 11973 topnfn 12164 iuncld 12323 txbas 12466 txdis 12485 xmetunirn 12566 xmettxlem 12717 xmettx 12718 pw1nct 13371 |
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