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Theorem rspceaov 41775
Description: A frequently used special case of rspc2ev 3455 for operation values, analogous to rspceov 6847. (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 2753 . . . 4 (𝑥 = 𝐶𝐹 = 𝐹)
2 id 22 . . . 4 (𝑥 = 𝐶𝑥 = 𝐶)
3 eqidd 2753 . . . 4 (𝑥 = 𝐶𝑦 = 𝑦)
41, 2, 3aoveq123d 41756 . . 3 (𝑥 = 𝐶 → ((𝑥𝐹𝑦)) = ((𝐶𝐹𝑦)) )
54eqeq2d 2762 . 2 (𝑥 = 𝐶 → (𝑆 = ((𝑥𝐹𝑦)) ↔ 𝑆 = ((𝐶𝐹𝑦)) ))
6 eqidd 2753 . . . 4 (𝑦 = 𝐷𝐹 = 𝐹)
7 eqidd 2753 . . . 4 (𝑦 = 𝐷𝐶 = 𝐶)
8 id 22 . . . 4 (𝑦 = 𝐷𝑦 = 𝐷)
96, 7, 8aoveq123d 41756 . . 3 (𝑦 = 𝐷 → ((𝐶𝐹𝑦)) = ((𝐶𝐹𝐷)) )
109eqeq2d 2762 . 2 (𝑦 = 𝐷 → (𝑆 = ((𝐶𝐹𝑦)) ↔ 𝑆 = ((𝐶𝐹𝐷)) ))
115, 10rspc2ev 3455 1 ((𝐶𝐴𝐷𝐵𝑆 = ((𝐶𝐹𝐷)) ) → ∃𝑥𝐴𝑦𝐵 𝑆 = ((𝑥𝐹𝑦)) )
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1624  wcel 2131  wrex 3043   ((caov 41693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-res 5270  df-iota 6004  df-fun 6043  df-fv 6049  df-dfat 41694  df-afv 41695  df-aov 41696
This theorem is referenced by: (None)
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