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Theorem rspc2ev 2658
Description: 2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.)
Hypotheses
Ref Expression
rspc2v.1 (x = A → (φχ))
rspc2v.2 (y = B → (χψ))
Assertion
Ref Expression
rspc2ev ((A 𝐶 B 𝐷 ψ) → x 𝐶 y 𝐷 φ)
Distinct variable groups:   x,y,A   y,B   x,𝐶   x,𝐷,y   χ,x   ψ,y
Allowed substitution hints:   φ(x,y)   ψ(x)   χ(y)   B(x)   𝐶(y)

Proof of Theorem rspc2ev
StepHypRef Expression
1 rspc2v.2 . . . . 5 (y = B → (χψ))
21rspcev 2650 . . . 4 ((B 𝐷 ψ) → y 𝐷 χ)
32anim2i 324 . . 3 ((A 𝐶 (B 𝐷 ψ)) → (A 𝐶 y 𝐷 χ))
433impb 1099 . 2 ((A 𝐶 B 𝐷 ψ) → (A 𝐶 y 𝐷 χ))
5 rspc2v.1 . . . 4 (x = A → (φχ))
65rexbidv 2321 . . 3 (x = A → (y 𝐷 φy 𝐷 χ))
76rspcev 2650 . 2 ((A 𝐶 y 𝐷 χ) → x 𝐶 y 𝐷 φ)
84, 7syl 14 1 ((A 𝐶 B 𝐷 ψ) → x 𝐶 y 𝐷 φ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  wrex 2301
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553
This theorem is referenced by:  rspc3ev  2660  opelxp  4317  rspceov  5489  2dom  6221  apreim  7367
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