Theorem r19.26-2 3147

Description: Restricted quantifier version of 19.26-2 1891. Version of r19.26 3122 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 3122	. . 3 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓))
2	1	ralbii 3108	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓))
3		r19.26 3122	. 2 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))
4	2, 3	bitri 277	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))

Syntax hints:   ↔ wb 208   ∧ wa 399  ∀wral 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829

This theorem depends on definitions:  df-bi 209  df-an 400  df-ral 3077

This theorem is referenced by:  fununi  6596  tz7.48lem  8412  isffth2  17951  ispos2  18347  issgrpv  18755  issgrpn0  18756  isnsg2  19197  efgred  19788  isrnghm  20486  dfrhm2  20519  df2idl2rng  21323  cpmatacl  22773  cpmatmcllem  22775  caucfil  25342  aalioulem6  26398  ajmoi  31058  adjmo  32032  prmidl2  33624  iccllysconn  35597  dfso3  36067  fvineqsnf1  37901  ispridl2  38534  disjimeceqim  39300  ishlat2  39974  fiinfi  44146  ntrk1k3eqk13  44623