Theorem bi2anan9 606

Description: Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 31-Jul-1995.)

Hypotheses

Ref	Expression
bi2an9.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
bi2an9.2	⊢ (𝜃 → (𝜏 ↔ 𝜂))

Assertion

Ref	Expression
bi2anan9	⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∧ 𝜏) ↔ (𝜒 ∧ 𝜂)))

Proof of Theorem bi2anan9

Step	Hyp	Ref	Expression
1		bi2an9.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	anbi1d 465	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜏) ↔ (𝜒 ∧ 𝜏)))
3		bi2an9.2	. . 3 ⊢ (𝜃 → (𝜏 ↔ 𝜂))
4	3	anbi2d 464	. 2 ⊢ (𝜃 → ((𝜒 ∧ 𝜏) ↔ (𝜒 ∧ 𝜂)))
5	2, 4	sylan9bb 462	1 ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∧ 𝜏) ↔ (𝜒 ∧ 𝜂)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bi2anan9r  607  rspc2gv  2880  ralprg  3674  raltpg  3676  prssg  3780  prsspwg  3783  opelopab2a  4300  opelxp  4694  eqrel  4753  eqrelrel  4765  brcog  4834  dff13  5818  resoprab2  6023  ovig  6048  dfoprab4f  6260  f1o2ndf1  6295  eroveu  6694  th3qlem1  6705  th3qlem2  6706  th3q  6708  oviec  6709  endisj  6892  exmidapne  7343  dfplpq2  7438  dfmpq2  7439  ordpipqqs  7458  enq0enq  7515  mulnnnq0  7534  ltsrprg  7831  axcnre  7965  axmulgt0  8115  addltmul  9245  ltxr  9867  sumsqeq0  10727  mul0inf  11423  dvds2lem  11985  opoe  12077  omoe  12078  opeo  12079  omeo  12080  gcddvds  12155  dfgcd2  12206  pcqmul  12497  xpsfrnel2  13048  eqgval  13429  txbasval  14587  cnmpt12  14607  cnmpt22  14614  lgsquadlem3  15404  lgsquad  15405  2sqlem7  15446