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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 8407 | . 2 ⊢ (𝐴 = 𝐵 → rec(𝐹, 𝐴) = rec(𝐹, 𝐵)) | |
2 | rdgeq1 8406 | . 2 ⊢ (𝐹 = 𝐺 → rec(𝐹, 𝐵) = rec(𝐺, 𝐵)) | |
3 | 1, 2 | sylan9eqr 2795 | 1 ⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐵) → rec(𝐹, 𝐴) = rec(𝐺, 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 reccrdg 8404 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-xp 5681 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-iota 6492 df-fv 6548 df-ov 7407 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 |
This theorem is referenced by: seqomeq12 8449 seqeq3 13967 satf 34282 satf0 34301 csbfinxpg 36207 |
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