Theorem bnj1101 30109

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

Hypotheses

Ref	Expression
bnj1101.1	⊢ ∃𝑥(𝜑 → 𝜓)
bnj1101.2	⊢ (𝜒 → 𝜑)

Assertion

Ref	Expression
bnj1101	⊢ ∃𝑥(𝜒 → 𝜓)

Proof of Theorem bnj1101

Step	Hyp	Ref	Expression
1		bnj1101.1	. . 3 ⊢ ∃𝑥(𝜑 → 𝜓)
2		pm3.42 581	. . 3 ⊢ ((𝜑 → 𝜓) → ((𝜒 ∧ 𝜑) → 𝜓))
3	1, 2	bnj101 30043	. 2 ⊢ ∃𝑥((𝜒 ∧ 𝜑) → 𝜓)
4		bnj1101.2	. . . . 5 ⊢ (𝜒 → 𝜑)
5	4	pm4.71i 662	. . . 4 ⊢ (𝜒 ↔ (𝜒 ∧ 𝜑))
6	5	imbi1i 338	. . 3 ⊢ ((𝜒 → 𝜓) ↔ ((𝜒 ∧ 𝜑) → 𝜓))
7	6	exbii 1764	. 2 ⊢ (∃𝑥(𝜒 → 𝜓) ↔ ∃𝑥((𝜒 ∧ 𝜑) → 𝜓))
8	3, 7	mpbir 220	1 ⊢ ∃𝑥(𝜒 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 383  ∃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:  bnj1110  30304  bnj1128  30312  bnj1145  30315