Theorem 19.35i 1803

Description: Inference associated with 19.35 1802. (Contributed by NM, 21-Jun-1993.)

Hypothesis

Ref	Expression
19.35i.1	⊢ ∃𝑥(𝜑 → 𝜓)

Assertion

Ref	Expression
19.35i	⊢ (∀𝑥𝜑 → ∃𝑥𝜓)

Proof of Theorem 19.35i

Step	Hyp	Ref	Expression
1		19.35i.1	. 2 ⊢ ∃𝑥(𝜑 → 𝜓)
2		19.35 1802	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
3	1, 2	mpbi 220	1 ⊢ (∀𝑥𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4  ∀wal 1478  ∃wex 1701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734

This theorem depends on definitions:  df-bi 197  df-ex 1702

This theorem is referenced by:  19.2  1889  spimeh  1922  cbv3hvOLD  2172  cbv3hvOLDOLD  2173  ax6e  2249  spimed  2254  equvini  2345  equvel  2346  euex  2493  axrep4  4745  zfcndrep  9396  bj-ax6elem2  32347  bj-spimedv  32414  bj-axrep4  32487  wl-exeq  32992  spd  41747