Theorem r19.36v 3181

Description: Restricted quantifier version of one direction of 19.36 2218. (The other direction holds iff 𝐴 is nonempty, see r19.36zv 4510.) (Contributed by NM, 22-Oct-2003.)

Assertion

Ref	Expression
r19.36v	⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → 𝜓))

Distinct variable group:   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem r19.36v

Step	Hyp	Ref	Expression
1		r19.35 3105	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
2		id 22	. . . 4 ⊢ (𝜓 → 𝜓)
3	2	rexlimivw 3148	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜓 → 𝜓)
4	3	imim2i 16	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → 𝜓))
5	1, 4	sylbi 216	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → 𝜓))

Syntax hints:   → wi 4  ∀wral 3058  ∃wrex 3067

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 395  df-ex 1774  df-ral 3059  df-rex 3068

This theorem is referenced by:  iinss  5063  uniimadom  10575  hashgt12el  14421

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