Theorem exlimiiv 1932

Description: Inference (Rule C) associated with exlimiv 1931. (Contributed by BJ, 19-Dec-2020.)

Hypotheses

Ref	Expression
exlimiv.1	⊢ (𝜑 → 𝜓)
exlimiiv.2	⊢ ∃𝑥𝜑

Assertion

Ref	Expression
exlimiiv	⊢ 𝜓

Distinct variable group:   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem exlimiiv

Step	Hyp	Ref	Expression
1		exlimiiv.2	. 2 ⊢ ∃𝑥𝜑
2		exlimiv.1	. . 3 ⊢ (𝜑 → 𝜓)
3	2	exlimiv 1931	. 2 ⊢ (∃𝑥𝜑 → 𝜓)
4	1, 3	ax-mp 5	1 ⊢ 𝜓

Syntax hints:   → wi 4  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911

This theorem depends on definitions:  df-bi 206  df-ex 1780

This theorem is referenced by:  equid  2013  ax7  2017  sbcom2  2159  ax12v2  2171  19.8a  2172  ax6e  2380  axc11n  2423  vtocl  3544  bm1.3ii  5301  axprlem1  5420  axprlem2  5421  axpr  5425  inf3  9632  epfrs  9728  kmlem2  10148  axcc2lem  10433  dcomex  10444  axdclem2  10517  grothpw  10823  grothpwex  10824  grothomex  10826  grothac  10827  cnso  16194  aannenlem3  26079  bj-ax12  35837  bj-ax6e  35848  wl-spae  36693