Theorem r19.26-2 3151

Description: Restricted quantifier version of 19.26-2 1904. Version of r19.26 3126 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 3126	. . 3 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓))
2	1	ralbii 3112	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓))
3		r19.26 3126	. 2 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))
4	2, 3	bitri 278	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∀wral 3080

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ral 3081

This theorem is used by:  fununi  6645  tz7.48lemOLD  8489  isffth2  18143  ispos2  18539  issgrpv  18960  issgrpn0  18961  isnsg2  19416  efgred  20013  dfring3  20569  isrnghm  20722  dfrhm2  20755  df2idl2rng  21601  prmidl2  21673  cpmatacl  23089  cpmatmcllem  23091  caucfil  25659  aalioulem6  26721  ajmoi  31517  adjmo  32491  iccllysconn  36223  dfso3  36700  fvineqsnf1  38601  ispridl2  39238  disjimeceqim  40002  ishlat2  40676  fiinfi  44835  ntrk1k3eqk13  45311