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Theorem rspceaov 39926
Description: A frequently used special case of rspc2ev 3295 for operation values, analogous to rspceov 6590. (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 2611 . . . 4 (𝑥 = 𝐶𝐹 = 𝐹)
2 id 22 . . . 4 (𝑥 = 𝐶𝑥 = 𝐶)
3 eqidd 2611 . . . 4 (𝑥 = 𝐶𝑦 = 𝑦)
41, 2, 3aoveq123d 39907 . . 3 (𝑥 = 𝐶 → ((𝑥𝐹𝑦)) = ((𝐶𝐹𝑦)) )
54eqeq2d 2620 . 2 (𝑥 = 𝐶 → (𝑆 = ((𝑥𝐹𝑦)) ↔ 𝑆 = ((𝐶𝐹𝑦)) ))
6 eqidd 2611 . . . 4 (𝑦 = 𝐷𝐹 = 𝐹)
7 eqidd 2611 . . . 4 (𝑦 = 𝐷𝐶 = 𝐶)
8 id 22 . . . 4 (𝑦 = 𝐷𝑦 = 𝐷)
96, 7, 8aoveq123d 39907 . . 3 (𝑦 = 𝐷 → ((𝐶𝐹𝑦)) = ((𝐶𝐹𝐷)) )
109eqeq2d 2620 . 2 (𝑦 = 𝐷 → (𝑆 = ((𝐶𝐹𝑦)) ↔ 𝑆 = ((𝐶𝐹𝐷)) ))
115, 10rspc2ev 3295 1 ((𝐶𝐴𝐷𝐵𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥𝐴𝑦𝐵 𝑆 = ((𝑥𝐹𝑦)) )
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1031   = wceq 1475  wcel 1977  wrex 2897   ((caov 39844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-dfat 39845  df-afv 39846  df-aov 39847
This theorem is referenced by: (None)
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