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Theorem rspceaov 44645
Description: A frequently used special case of rspc2ev 3572 for operation values, analogous to rspceov 7315. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
rspceaov ((𝐶𝐴𝐷𝐵𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥𝐴𝑦𝐵 𝑆 = ((𝑥𝐹𝑦)) )
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦   𝑦,𝐷   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐴(𝑦)   𝐷(𝑥)

Proof of Theorem rspceaov
StepHypRef Expression
1 eqidd 2739 . . . 4 (𝑥 = 𝐶𝐹 = 𝐹)
2 id 22 . . . 4 (𝑥 = 𝐶𝑥 = 𝐶)
3 eqidd 2739 . . . 4 (𝑥 = 𝐶𝑦 = 𝑦)
41, 2, 3aoveq123d 44626 . . 3 (𝑥 = 𝐶 → ((𝑥𝐹𝑦)) = ((𝐶𝐹𝑦)) )
54eqeq2d 2749 . 2 (𝑥 = 𝐶 → (𝑆 = ((𝑥𝐹𝑦)) ↔ 𝑆 = ((𝐶𝐹𝑦)) ))
6 eqidd 2739 . . . 4 (𝑦 = 𝐷𝐹 = 𝐹)
7 eqidd 2739 . . . 4 (𝑦 = 𝐷𝐶 = 𝐶)
8 id 22 . . . 4 (𝑦 = 𝐷𝑦 = 𝐷)
96, 7, 8aoveq123d 44626 . . 3 (𝑦 = 𝐷 → ((𝐶𝐹𝑦)) = ((𝐶𝐹𝐷)) )
109eqeq2d 2749 . 2 (𝑦 = 𝐷 → (𝑆 = ((𝐶𝐹𝑦)) ↔ 𝑆 = ((𝐶𝐹𝐷)) ))
115, 10rspc2ev 3572 1 ((𝐶𝐴𝐷𝐵𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥𝐴𝑦𝐵 𝑆 = ((𝑥𝐹𝑦)) )
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1539  wcel 2106  wrex 3065   ((caov 44566
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pr 5351
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3432  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-int 4881  df-br 5075  df-opab 5137  df-id 5485  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-res 5597  df-iota 6385  df-fun 6429  df-fv 6435  df-aiota 44533  df-dfat 44567  df-afv 44568  df-aov 44569
This theorem is referenced by: (None)
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