Theorem bnj1023 30105

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1023.1	⊢ ∃𝑥(𝜑 → 𝜓)
bnj1023.2	⊢ (𝜓 → 𝜒)

Assertion

Ref	Expression
bnj1023	⊢ ∃𝑥(𝜑 → 𝜒)

Proof of Theorem bnj1023

Step	Hyp	Ref	Expression
1		bnj1023.2	. . . . 5 ⊢ (𝜓 → 𝜒)
2	1	a1i 11	. . . 4 ⊢ ((𝜑 → 𝜓) → (𝜓 → 𝜒))
3	2	ax-gen 1713	. . 3 ⊢ ∀𝑥((𝜑 → 𝜓) → (𝜓 → 𝜒))
4		bnj1023.1	. . 3 ⊢ ∃𝑥(𝜑 → 𝜓)
5		exintr 1810	. . 3 ⊢ (∀𝑥((𝜑 → 𝜓) → (𝜓 → 𝜒)) → (∃𝑥(𝜑 → 𝜓) → ∃𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜒))))
6	3, 4, 5	mp2 9	. 2 ⊢ ∃𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜒))
7		pm3.33 607	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜒)) → (𝜑 → 𝜒))
8	6, 7	bnj101 30043	1 ⊢ ∃𝑥(𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by:  bnj1098  30108  bnj1110  30304  bnj1118  30306  bnj1128  30312  bnj1145  30315