Theorem bnj1209 34335

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

Hypotheses

Ref	Expression
bnj1209.1	⊢ (𝜒 → ∃𝑥 ∈ 𝐵 𝜑)
bnj1209.2	⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑))

Assertion

Ref	Expression
bnj1209	⊢ (𝜒 → ∃𝑥𝜃)

Distinct variable group:   𝜒,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜃(𝑥)   𝐵(𝑥)

Proof of Theorem bnj1209

Step	Hyp	Ref	Expression
1		bnj1209.1	. . . . 5 ⊢ (𝜒 → ∃𝑥 ∈ 𝐵 𝜑)
2	1	bnj1196 34333	. . . 4 ⊢ (𝜒 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
3	2	ancli 548	. . 3 ⊢ (𝜒 → (𝜒 ∧ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
4		19.42v 1949	. . 3 ⊢ (∃𝑥(𝜒 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)) ↔ (𝜒 ∧ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
5	3, 4	sylibr 233	. 2 ⊢ (𝜒 → ∃𝑥(𝜒 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)))
6		bnj1209.2	. . 3 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑))
7		3anass 1092	. . 3 ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝜒 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)))
8	6, 7	bitri 275	. 2 ⊢ (𝜃 ↔ (𝜒 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)))
9	5, 8	bnj1198 34334	1 ⊢ (𝜒 → ∃𝑥𝜃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084  ∃wex 1773   ∈ wcel 2098  ∃wrex 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-ex 1774  df-rex 3065

This theorem is referenced by:  bnj1501  34606  bnj1523  34610