Theorem r19.28av 2449

Description: Restricted version of one direction of Theorem 19.28 of [Margaris] p. 90. (The other direction doesn't hold when 𝐴 is empty.) (Contributed by NM, 2-Apr-2004.)

Assertion

Ref	Expression
r19.28av	⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem r19.28av

Step	Hyp	Ref	Expression
1		r19.27av 2448	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜓 ∧ 𝜑) → ∀𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑))
2		ancom 253	. 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜓 ∧ 𝜑))
3		ancom 253	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
4	3	ralbii 2330	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑))
5	1, 2, 4	3imtr4i 190	1 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 97  ∀wral 2306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400  ax-17 1419

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-ral 2311

This theorem is referenced by:  rr19.28v  2683  fununi  4967