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Mirrors > Home > ILE Home > Th. List > iftruei | GIF version |
Description: Inference associated with iftrue 3539. (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 3539 | . 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 1353 ifcif 3534 |
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-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-if 3535 |
This theorem is referenced by: ctmlemr 7106 xnegpnf 9826 xnegmnf 9827 xaddpnf1 9844 xaddpnf2 9845 xaddmnf1 9846 xaddmnf2 9847 pnfaddmnf 9848 mnfaddpnf 9849 iseqf1olemqk 10491 exp0 10521 sumsnf 11412 prodsnf 11595 lcm0val 12059 ennnfonelemj0 12396 ennnfonelem0 12400 mulg0 12942 lgs0 14307 lgs2 14311 peano3nninf 14638 dceqnconst 14689 |
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