Theorem bnj31 32410

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

Hypotheses

Ref	Expression
bnj31.1	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
bnj31.2	⊢ (𝜓 → 𝜒)

Assertion

Ref	Expression
bnj31	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒)

Proof of Theorem bnj31

Step	Hyp	Ref	Expression
1		bnj31.1	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
2		bnj31.2	. . 3 ⊢ (𝜓 → 𝜒)
3	2	reximi 3166	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒)
4	1, 3	syl 17	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒)

Syntax hints:   → wi 4  ∃wrex 3062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-ral 3066  df-rex 3067

This theorem is referenced by:  bnj168  32421  bnj110  32551  bnj906  32623  bnj1253  32710  bnj1280  32713  bnj1296  32714  bnj1371  32722  bnj1497  32753  bnj1498  32754  bnj1501  32760