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  5229  funco  5358  fliftval  5930  ltsrprg  7942  difelfznle  10339  nelfzo  10356  iseqf1olemqk  10737  ccatsymb  11145  pfxsuffeqwrdeq  11238  pfxccatin12lem2a  11267  difsqpwdvds  12869  resmhm  13528  mhmco  13531  rhmco  14146  resrhm  14220  gausslemma2dlem1a  15745  ex-ceil  16114