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Mirrors > Home > MPE Home > Th. List > ifeq2da | Structured version Visualization version GIF version |
Description: Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
ifeq2da.1 | ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
ifeq2da | ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 4419 | . . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐴) = 𝐶) | |
2 | iftrue 4419 | . . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐵) = 𝐶) | |
3 | 1, 2 | eqtr4d 2797 | . . 3 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) |
4 | 3 | adantl 486 | . 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) |
5 | ifeq2da.1 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵) | |
6 | 5 | ifeq2d 4433 | . 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) |
7 | 4, 6 | pm2.61dan 813 | 1 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1539 ifcif 4413 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-12 2176 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-rab 3077 df-v 3409 df-un 3859 df-if 4414 |
This theorem is referenced by: ifeq12da 4446 dfac12lem1 9588 ttukeylem3 9956 xmulcom 12685 xmulneg1 12688 subgmulg 18345 1marepvmarrepid 21260 pcopt2 23709 limcdif 24560 limcmpt 24567 limcres 24570 limccnp 24575 radcnv0 25095 leibpi 25612 efrlim 25639 dchrvmasumiflem2 26170 padicabvf 26299 padicabvcxp 26300 itg2addnclem 35373 fourierdlem73 43172 fourierdlem76 43175 fourierdlem89 43188 fourierdlem91 43190 |
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