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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 2689), 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 2689 | . 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 1480 Vcvv 2686 |
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 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-v 2688 |
This theorem is referenced by: xpiindim 4676 disjxp1 6133 ixpiinm 6618 ixpsnf1o 6630 iunfidisj 6834 ssfii 6862 fifo 6868 omp1eomlem 6979 exmidomniim 7013 bcval5 10514 rexfiuz 10766 fsum2dlemstep 11208 fsumcnv 11211 fisumcom2 11212 fsumconst 11228 modfsummodlemstep 11231 fsumabs 11239 ennnfonelemim 11942 topnfn 12130 iuncld 12289 txbas 12432 txdis 12451 xmetunirn 12532 xmettxlem 12683 xmettx 12684 |
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