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Theorem rspceaov 45503
Description: A frequently used special case of rspc2ev 3595 for operation values, analogous to rspceov 7409. (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 2738 . . . 4 (𝑥 = 𝐶𝐹 = 𝐹)
2 id 22 . . . 4 (𝑥 = 𝐶𝑥 = 𝐶)
3 eqidd 2738 . . . 4 (𝑥 = 𝐶𝑦 = 𝑦)
41, 2, 3aoveq123d 45484 . . 3 (𝑥 = 𝐶 → ((𝑥𝐹𝑦)) = ((𝐶𝐹𝑦)) )
54eqeq2d 2748 . 2 (𝑥 = 𝐶 → (𝑆 = ((𝑥𝐹𝑦)) ↔ 𝑆 = ((𝐶𝐹𝑦)) ))
6 eqidd 2738 . . . 4 (𝑦 = 𝐷𝐹 = 𝐹)
7 eqidd 2738 . . . 4 (𝑦 = 𝐷𝐶 = 𝐶)
8 id 22 . . . 4 (𝑦 = 𝐷𝑦 = 𝐷)
96, 7, 8aoveq123d 45484 . . 3 (𝑦 = 𝐷 → ((𝐶𝐹𝑦)) = ((𝐶𝐹𝐷)) )
109eqeq2d 2748 . 2 (𝑦 = 𝐷 → (𝑆 = ((𝐶𝐹𝑦)) ↔ 𝑆 = ((𝐶𝐹𝐷)) ))
115, 10rspc2ev 3595 1 ((𝐶𝐴𝐷𝐵𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥𝐴𝑦𝐵 𝑆 = ((𝑥𝐹𝑦)) )
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2107  wrex 3074   ((caov 45424
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
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-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-res 5650  df-iota 6453  df-fun 6503  df-fv 6509  df-aiota 45391  df-dfat 45425  df-afv 45426  df-aov 45427
This theorem is referenced by: (None)
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