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Mirrors > Home > ILE Home > Th. List > iftruei | GIF version |
Description: Inference associated with iftrue 3474. (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 3474 | . 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 1331 ifcif 3469 |
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 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-if 3470 |
This theorem is referenced by: ctmlemr 6986 xnegpnf 9604 xnegmnf 9605 xaddpnf1 9622 xaddpnf2 9623 xaddmnf1 9624 xaddmnf2 9625 pnfaddmnf 9626 mnfaddpnf 9627 iseqf1olemqk 10260 exp0 10290 sumsnf 11171 lcm0val 11735 ennnfonelemj0 11903 ennnfonelem0 11907 peano3nninf 13190 |
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