Theorem ssoprab2 6909

Description: Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 5162. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

Assertion

Ref	Expression
ssoprab2	⊢ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓})

Proof of Theorem ssoprab2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . . . 7 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓))
2	1	anim2d 605	. . . . . 6 ⊢ ((𝜑 → 𝜓) → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)))
3	2	aleximi 1926	. . . . 5 ⊢ (∀𝑧(𝜑 → 𝜓) → (∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)))
4	3	aleximi 1926	. . . 4 ⊢ (∀𝑦∀𝑧(𝜑 → 𝜓) → (∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)))
5	4	aleximi 1926	. . 3 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)))
6	5	ss2abdv 3835	. 2 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) → {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} ⊆ {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)})
7		df-oprab 6846	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
8		df-oprab 6846	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)}
9	6, 7, 8	3sstr4g 3806	1 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓})

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1650   = wceq 1652  ∃wex 1874  {cab 2751   ⊆ wss 3732  ⟨cop 4340  {coprab 6843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-in 3739  df-ss 3746  df-oprab 6846

This theorem is referenced by:  ssoprab2b  6910  joinfval  17269  meetfval  17283