Theorem ssoprab2 7479

Description: Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 5532. (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 23	. . . . . . 7 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓))
2	1	anim2d 623	. . . . . 6 ⊢ ((𝜑 → 𝜓) → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)))
3	2	aleximi 1859	. . . . 5 ⊢ (∀𝑧(𝜑 → 𝜓) → (∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)))
4	3	aleximi 1859	. . . 4 ⊢ (∀𝑦∀𝑧(𝜑 → 𝜓) → (∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)))
5	4	aleximi 1859	. . 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 4027	. 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 7415	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
8		df-oprab 7415	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)}
9	6, 7, 8	3sstr4g 3998	1 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓})

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1565   = wceq 1567  ∃wex 1806  {cab 2747   ⊆ wss 3913  ⟨cop 4600  {coprab 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-ss 3930  df-oprab 7415

This theorem is referenced by:  ssoprab2b  7480  eqoprab2bw  7481  joinfval  18426  meetfval  18440