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Theorem rspceaov 41783
Description: A frequently used special case of rspc2ev 3463 for operation values, analogous to rspceov 6855. (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 2761 . . . 4 (𝑥 = 𝐶𝐹 = 𝐹)
2 id 22 . . . 4 (𝑥 = 𝐶𝑥 = 𝐶)
3 eqidd 2761 . . . 4 (𝑥 = 𝐶𝑦 = 𝑦)
41, 2, 3aoveq123d 41764 . . 3 (𝑥 = 𝐶 → ((𝑥𝐹𝑦)) = ((𝐶𝐹𝑦)) )
54eqeq2d 2770 . 2 (𝑥 = 𝐶 → (𝑆 = ((𝑥𝐹𝑦)) ↔ 𝑆 = ((𝐶𝐹𝑦)) ))
6 eqidd 2761 . . . 4 (𝑦 = 𝐷𝐹 = 𝐹)
7 eqidd 2761 . . . 4 (𝑦 = 𝐷𝐶 = 𝐶)
8 id 22 . . . 4 (𝑦 = 𝐷𝑦 = 𝐷)
96, 7, 8aoveq123d 41764 . . 3 (𝑦 = 𝐷 → ((𝐶𝐹𝑦)) = ((𝐶𝐹𝐷)) )
109eqeq2d 2770 . 2 (𝑦 = 𝐷 → (𝑆 = ((𝐶𝐹𝑦)) ↔ 𝑆 = ((𝐶𝐹𝐷)) ))
115, 10rspc2ev 3463 1 ((𝐶𝐴𝐷𝐵𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥𝐴𝑦𝐵 𝑆 = ((𝑥𝐹𝑦)) )
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1632  wcel 2139  wrex 3051   ((caov 41701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-res 5278  df-iota 6012  df-fun 6051  df-fv 6057  df-dfat 41702  df-afv 41703  df-aov 41704
This theorem is referenced by: (None)
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