Theorem r19.21biOLD 3208

Description: Obsolete version of r19.21bi 3207 as of 11-Jun-2023. (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 1-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
r19.21bi.1	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Assertion

Ref	Expression
r19.21biOLD	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)

Proof of Theorem r19.21biOLD

Step	Hyp	Ref	Expression
1		r19.21bi.1	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
2		rsp 3204	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 → (𝑥 ∈ 𝐴 → 𝜓))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))
4	3	imp 409	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2113  ∀wral 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-ral 3142

This theorem is referenced by: (None)