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Mirrors > Home > ILE Home > Th. List > iftruei | GIF version |
Description: Inference associated with iftrue 3484. (Contributed by BJ, 7-Oct-2018.) |
Ref | Expression |
---|---|
iftruei.1 | ⊢ 𝜑 |
Ref | Expression |
---|---|
iftruei | ⊢ if(𝜑, 𝐴, 𝐵) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftruei.1 | . 2 ⊢ 𝜑 | |
2 | iftrue 3484 | . 2 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ifcif 3479 |
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-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-11 1485 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-if 3480 |
This theorem is referenced by: ctmlemr 7001 xnegpnf 9641 xnegmnf 9642 xaddpnf1 9659 xaddpnf2 9660 xaddmnf1 9661 xaddmnf2 9662 pnfaddmnf 9663 mnfaddpnf 9664 iseqf1olemqk 10298 exp0 10328 sumsnf 11210 lcm0val 11782 ennnfonelemj0 11950 ennnfonelem0 11954 peano3nninf 13376 dceqnconst 13423 |
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