Theorem 19.35i 1911

Description: Inference associated with 19.35 1910. (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 1910	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
3	1, 2	mpbi 233	1 ⊢ (∀𝑥𝜑 → ∃𝑥𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  19.2  2009  spimedv  2236  ax6e  2418  spimed  2423  equvini  2490  equvel  2491  axrep4OLD  5250  zfcndrep  10617  bj-ax6elem2  37330  wl-exeq  38230  spd  50497