Theorem 2eximdv 1917

Description: Deduction form of Theorem 19.22 of [Margaris] p. 90 with two quantifiers, see exim 1831. (Contributed by NM, 3-Aug-1995.)

Hypothesis

Ref	Expression
2alimdv.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
2eximdv	⊢ (𝜑 → (∃𝑥∃𝑦𝜓 → ∃𝑥∃𝑦𝜒))

Distinct variable groups:   𝜑,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem 2eximdv

Step	Hyp	Ref	Expression
1		2alimdv.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	eximdv 1915	. 2 ⊢ (𝜑 → (∃𝑦𝜓 → ∃𝑦𝜒))
3	2	eximdv 1915	1 ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 → ∃𝑥∃𝑦𝜒))

Syntax hints:   → wi 4  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908

This theorem depends on definitions:  df-bi 207  df-ex 1777

This theorem is referenced by:  2eu6  2655  cgsex2g  3525  cgsex4g  3526  cgsex4gOLD  3527  spc2egv  3599  rexopabb  5538  relop  5864  elinxp  6039  opreuopreu  8058  en3  9314  en4  9315  addsrpr  11113  mulsrpr  11114  hash2prde  14506  hash3tpde  14529  pmtrrn2  19493  umgredg  29170  umgr2wlkon  29980  trsp2cyc  33126  acycgrsubgr  35143  satfvsucsuc  35350  fmla0xp  35368  fundmpss  35748  pellexlem5  42821  ax6e2eq  44555  fnchoice  44967  fzisoeu  45251  stoweidlem35  45991  stoweidlem60  46016  or2expropbi  46984  ich2exprop  47396