Theorem ssoprab2 6134

Description: Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4413. (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 19	. . . . . . . . . 10 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓))
2	1	anim2d 337	. . . . . . . . 9 ⊢ ((𝜑 → 𝜓) → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)))
3	2	alimi 1508	. . . . . . . 8 ⊢ (∀𝑧(𝜑 → 𝜓) → ∀𝑧((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)))
4		exim 1652	. . . . . . . 8 ⊢ (∀𝑧((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)) → (∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)))
5	3, 4	syl 14	. . . . . . 7 ⊢ (∀𝑧(𝜑 → 𝜓) → (∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)))
6	5	alimi 1508	. . . . . 6 ⊢ (∀𝑦∀𝑧(𝜑 → 𝜓) → ∀𝑦(∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)))
7		exim 1652	. . . . . 6 ⊢ (∀𝑦(∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)) → (∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)))
8	6, 7	syl 14	. . . . 5 ⊢ (∀𝑦∀𝑧(𝜑 → 𝜓) → (∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)))
9	8	alimi 1508	. . . 4 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) → ∀𝑥(∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)))
10		exim 1652	. . . 4 ⊢ (∀𝑥(∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)) → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)))
11	9, 10	syl 14	. . 3 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)))
12	11	ss2abdv 3321	. 2 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) → {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} ⊆ {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)})
13		df-oprab 6079	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
14		df-oprab 6079	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)}
15	12, 13, 14	3sstr4g 3291	1 ⊢ (∀𝑥∀𝑦∀𝑧(𝜑 → 𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓})

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1400   = wceq 1402  ∃wex 1545  {cab 2224   ⊆ wss 3220  ⟨cop 3708  {coprab 6076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-in 3226  df-ss 3233  df-oprab 6079

This theorem is referenced by:  ssoprab2b  6135