Theorem anim12ci 339

Description: Variant of anim12i 338 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
anim12i.1	⊢ (𝜑 → 𝜓)
anim12i.2	⊢ (𝜒 → 𝜃)

Assertion

Ref	Expression
anim12ci	⊢ ((𝜑 ∧ 𝜒) → (𝜃 ∧ 𝜓))

Proof of Theorem anim12ci

Step	Hyp	Ref	Expression
1		anim12i.2	. . 3 ⊢ (𝜒 → 𝜃)
2		anim12i.1	. . 3 ⊢ (𝜑 → 𝜓)
3	1, 2	anim12i 338	. 2 ⊢ ((𝜒 ∧ 𝜑) → (𝜃 ∧ 𝜓))
4	3	ancoms 268	1 ⊢ ((𝜑 ∧ 𝜒) → (𝜃 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 104

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 is referenced by:  anim1ci  341  dfco2a  5286  funco  5415  fliftval  6000  ltsrprg  8108  difelfznle  10525  nelfzo  10542  iseqf1olemqk  10927  ccatsymb  11353  pfxsuffeqwrdeq  11453  pfxccatin12lem2a  11482  difsqpwdvds  13100  resmhm  13777  mhmco  13780  rhmco  14464  resrhm  14539  gausslemma2dlem1a  16160  subusgr  16499  ex-ceil  16723