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Mirrors > Home > MPE Home > Th. List > rdgeq12 | Structured version Visualization version GIF version |
Description: Equality theorem for the recursive definition generator. (Contributed by Scott Fenton, 28-Apr-2012.) |
Ref | Expression |
---|---|
rdgeq12 | ⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐵) → rec(𝐹, 𝐴) = rec(𝐺, 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgeq2 8050 | . 2 ⊢ (𝐴 = 𝐵 → rec(𝐹, 𝐴) = rec(𝐹, 𝐵)) | |
2 | rdgeq1 8049 | . 2 ⊢ (𝐹 = 𝐺 → rec(𝐹, 𝐵) = rec(𝐺, 𝐵)) | |
3 | 1, 2 | sylan9eqr 2880 | 1 ⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐵) → rec(𝐹, 𝐴) = rec(𝐺, 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 reccrdg 8047 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-xp 5563 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-iota 6316 df-fv 6365 df-wrecs 7949 df-recs 8010 df-rdg 8048 |
This theorem is referenced by: seqomeq12 8092 seqeq3 13377 satf 32602 satf0 32621 trpredeq1 33061 trpredeq2 33062 trpred0 33077 csbfinxpg 34671 |
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