Theorem rspc2ev 2963

Description: 2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.)

Hypotheses

Ref	Expression
rspc2v.1	⊢ (x = A → (φ ↔ χ))
rspc2v.2	⊢ (y = B → (χ ↔ ψ))

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

Step	Hyp	Ref	Expression
1		rspc2v.2	. . . . 5 ⊢ (y = B → (χ ↔ ψ))
2	1	rspcev 2955	. . . 4 ⊢ ((B ∈ D ∧ ψ) → ∃y ∈ D χ)
3	2	anim2i 552	. . 3 ⊢ ((A ∈ C ∧ (B ∈ D ∧ ψ)) → (A ∈ C ∧ ∃y ∈ D χ))
4	3	3impb 1147	. 2 ⊢ ((A ∈ C ∧ B ∈ D ∧ ψ) → (A ∈ C ∧ ∃y ∈ D χ))
5		rspc2v.1	. . . 4 ⊢ (x = A → (φ ↔ χ))
6	5	rexbidv 2635	. . 3 ⊢ (x = A → (∃y ∈ D φ ↔ ∃y ∈ D χ))
7	6	rspcev 2955	. 2 ⊢ ((A ∈ C ∧ ∃y ∈ D χ) → ∃x ∈ C ∃y ∈ D φ)
8	4, 7	syl 15	1 ⊢ ((A ∈ C ∧ B ∈ D ∧ ψ) → ∃x ∈ C ∃y ∈ D φ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∃wrex 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-3an 936  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:  rspc3ev  2965  eladdci  4399  rspceov  5556  nclec  6195  ltcpw1pwg  6202  nc0le1  6216  nclenc  6222  ce2le  6233  tlenc1c  6240