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Theorem ceqsrex2v 2974
 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 630 . . . . . 6 (((x = A y = B) φ) ↔ (x = A (y = B φ)))
21rexbii 2639 . . . . 5 (y D ((x = A y = B) φ) ↔ y D (x = A (y = B φ)))
3 r19.42v 2765 . . . . 5 (y D (x = A (y = B φ)) ↔ (x = A y D (y = B φ)))
42, 3bitri 240 . . . 4 (y D ((x = A y = B) φ) ↔ (x = A y D (y = B φ)))
54rexbii 2639 . . 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 684 . . . . 5 (x = A → ((y = B φ) ↔ (y = B ψ)))
87rexbidv 2635 . . . 4 (x = A → (y D (y = B φ) ↔ y D (y = B ψ)))
98ceqsrexv 2972 . . 3 (A C → (x C (x = A y D (y = B φ)) ↔ y D (y = B ψ)))
105, 9syl5bb 248 . 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 2972 . 2 (B D → (y D (y = B ψ) ↔ χ))
1310, 12sylan9bb 680 1 ((A C B D) → (x C y D ((x = A y = B) φ) ↔ χ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861 This theorem is referenced by: (None)
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