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Theorem ceqsrex2v 2670
Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.)
Hypotheses
Ref Expression
ceqsrex2v.1 (x = A → (φψ))
ceqsrex2v.2 (y = B → (ψχ))
Assertion
Ref Expression
ceqsrex2v ((A 𝐶 B 𝐷) → (x 𝐶 y 𝐷 ((x = A y = B) φ) ↔ χ))
Distinct variable groups:   x,y,A   x,B,y   x,𝐶   x,𝐷,y   ψ,x   χ,y
Allowed substitution hints:   φ(x,y)   ψ(y)   χ(x)   𝐶(y)

Proof of Theorem ceqsrex2v
StepHypRef Expression
1 anass 381 . . . . . 6 (((x = A y = B) φ) ↔ (x = A (y = B φ)))
21rexbii 2325 . . . . 5 (y 𝐷 ((x = A y = B) φ) ↔ y 𝐷 (x = A (y = B φ)))
3 r19.42v 2461 . . . . 5 (y 𝐷 (x = A (y = B φ)) ↔ (x = A y 𝐷 (y = B φ)))
42, 3bitri 173 . . . 4 (y 𝐷 ((x = A y = B) φ) ↔ (x = A y 𝐷 (y = B φ)))
54rexbii 2325 . . 3 (x 𝐶 y 𝐷 ((x = A y = B) φ) ↔ x 𝐶 (x = A y 𝐷 (y = B φ)))
6 ceqsrex2v.1 . . . . . 6 (x = A → (φψ))
76anbi2d 437 . . . . 5 (x = A → ((y = B φ) ↔ (y = B ψ)))
87rexbidv 2321 . . . 4 (x = A → (y 𝐷 (y = B φ) ↔ y 𝐷 (y = B ψ)))
98ceqsrexv 2668 . . 3 (A 𝐶 → (x 𝐶 (x = A y 𝐷 (y = B φ)) ↔ y 𝐷 (y = B ψ)))
105, 9syl5bb 181 . 2 (A 𝐶 → (x 𝐶 y 𝐷 ((x = A y = B) φ) ↔ y 𝐷 (y = B ψ)))
11 ceqsrex2v.2 . . 3 (y = B → (ψχ))
1211ceqsrexv 2668 . 2 (B 𝐷 → (y 𝐷 (y = B ψ) ↔ χ))
1310, 12sylan9bb 435 1 ((A 𝐶 B 𝐷) → (x 𝐶 y 𝐷 ((x = A y = B) φ) ↔ χ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = 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-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: (None)
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