Theorem r19.26-2 3065

Description: Restricted quantifier version of 19.26-2 1799. Version of r19.26 3064 with two quantifiers. (Contributed by NM, 10-Aug-2004.)

Assertion

Ref	Expression
r19.26-2	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))

Proof of Theorem r19.26-2

Step	Hyp	Ref	Expression
1		r19.26 3064	. . 3 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓))
2	1	ralbii 2980	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓))
3		r19.26 3064	. 2 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))
4	2, 3	bitri 264	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))

Syntax hints:   ↔ wb 196   ∧ wa 384  ∀wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386  df-ral 2917

This theorem is referenced by:  fununi  5964  tz7.48lem  7536  isffth2  16576  ispos2  16948  issgrpv  17286  issgrpn0  17287  isnsg2  17624  efgred  18161  dfrhm2  18717  cpmatacl  20521  cpmatmcllem  20523  caucfil  23081  aalioulem6  24092  ajmoi  27714  adjmo  28691  iccllysconn  31232  dfso3  31601  ispridl2  33837  ishlat2  34640  fiinfi  37878  ntrk1k3eqk13  38348  isrnghm  41892