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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 8359 | . 2 ⊢ (𝐴 = 𝐵 → rec(𝐹, 𝐴) = rec(𝐹, 𝐵)) | |
2 | rdgeq1 8358 | . 2 ⊢ (𝐹 = 𝐺 → rec(𝐹, 𝐵) = rec(𝐺, 𝐵)) | |
3 | 1, 2 | sylan9eqr 2799 | 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 8356 |
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 2708 |
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 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-xp 5640 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-iota 6449 df-fv 6505 df-ov 7361 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 |
This theorem is referenced by: seqomeq12 8401 seqeq3 13912 satf 33950 satf0 33969 csbfinxpg 35862 |
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