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Mirrors > Home > NFE Home > Th. List > ssopab2 | GIF version |
Description: Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 19-May-2013.) |
Ref | Expression |
---|---|
ssopab2 | ⊢ (∀x∀y(φ → ψ) → {〈x, y〉 ∣ φ} ⊆ {〈x, y〉 ∣ ψ}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 1788 | . . . 4 ⊢ Ⅎx∀x∀y(φ → ψ) | |
2 | nfa1 1788 | . . . . . 6 ⊢ Ⅎy∀y(φ → ψ) | |
3 | sp 1747 | . . . . . . 7 ⊢ (∀y(φ → ψ) → (φ → ψ)) | |
4 | 3 | anim2d 548 | . . . . . 6 ⊢ (∀y(φ → ψ) → ((z = 〈x, y〉 ∧ φ) → (z = 〈x, y〉 ∧ ψ))) |
5 | 2, 4 | eximd 1770 | . . . . 5 ⊢ (∀y(φ → ψ) → (∃y(z = 〈x, y〉 ∧ φ) → ∃y(z = 〈x, y〉 ∧ ψ))) |
6 | 5 | sps 1754 | . . . 4 ⊢ (∀x∀y(φ → ψ) → (∃y(z = 〈x, y〉 ∧ φ) → ∃y(z = 〈x, y〉 ∧ ψ))) |
7 | 1, 6 | eximd 1770 | . . 3 ⊢ (∀x∀y(φ → ψ) → (∃x∃y(z = 〈x, y〉 ∧ φ) → ∃x∃y(z = 〈x, y〉 ∧ ψ))) |
8 | 7 | ss2abdv 3339 | . 2 ⊢ (∀x∀y(φ → ψ) → {z ∣ ∃x∃y(z = 〈x, y〉 ∧ φ)} ⊆ {z ∣ ∃x∃y(z = 〈x, y〉 ∧ ψ)}) |
9 | df-opab 4623 | . 2 ⊢ {〈x, y〉 ∣ φ} = {z ∣ ∃x∃y(z = 〈x, y〉 ∧ φ)} | |
10 | df-opab 4623 | . 2 ⊢ {〈x, y〉 ∣ ψ} = {z ∣ ∃x∃y(z = 〈x, y〉 ∧ ψ)} | |
11 | 8, 9, 10 | 3sstr4g 3312 | 1 ⊢ (∀x∀y(φ → ψ) → {〈x, y〉 ∣ φ} ⊆ {〈x, y〉 ∣ ψ}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∀wal 1540 ∃wex 1541 = wceq 1642 {cab 2339 ⊆ wss 3257 〈cop 4561 {copab 4622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-opab 4623 |
This theorem is referenced by: ssopab2b 4713 ssopab2i 4714 ssopab2dv 4715 |
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