Theorem ssopab2b 4714

Description: Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Assertion

Ref	Expression
ssopab2b	⊢ ({⟨x, y⟩ ∣ φ} ⊆ {⟨x, y⟩ ∣ ψ} ↔ ∀x∀y(φ → ψ))

Proof of Theorem ssopab2b

Step	Hyp	Ref	Expression
1		nfopab1 4629	. . . 4 ⊢ Ⅎx{⟨x, y⟩ ∣ φ}
2		nfopab1 4629	. . . 4 ⊢ Ⅎx{⟨x, y⟩ ∣ ψ}
3	1, 2	nfss 3267	. . 3 ⊢ Ⅎx{⟨x, y⟩ ∣ φ} ⊆ {⟨x, y⟩ ∣ ψ}
4		nfopab2 4630	. . . . 5 ⊢ Ⅎy{⟨x, y⟩ ∣ φ}
5		nfopab2 4630	. . . . 5 ⊢ Ⅎy{⟨x, y⟩ ∣ ψ}
6	4, 5	nfss 3267	. . . 4 ⊢ Ⅎy{⟨x, y⟩ ∣ φ} ⊆ {⟨x, y⟩ ∣ ψ}
7		ssel 3268	. . . . 5 ⊢ ({⟨x, y⟩ ∣ φ} ⊆ {⟨x, y⟩ ∣ ψ} → (⟨x, y⟩ ∈ {⟨x, y⟩ ∣ φ} → ⟨x, y⟩ ∈ {⟨x, y⟩ ∣ ψ}))
8		opabid 4696	. . . . 5 ⊢ (⟨x, y⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ φ)
9		opabid 4696	. . . . 5 ⊢ (⟨x, y⟩ ∈ {⟨x, y⟩ ∣ ψ} ↔ ψ)
10	7, 8, 9	3imtr3g 260	. . . 4 ⊢ ({⟨x, y⟩ ∣ φ} ⊆ {⟨x, y⟩ ∣ ψ} → (φ → ψ))
11	6, 10	alrimi 1765	. . 3 ⊢ ({⟨x, y⟩ ∣ φ} ⊆ {⟨x, y⟩ ∣ ψ} → ∀y(φ → ψ))
12	3, 11	alrimi 1765	. 2 ⊢ ({⟨x, y⟩ ∣ φ} ⊆ {⟨x, y⟩ ∣ ψ} → ∀x∀y(φ → ψ))
13		ssopab2 4713	. 2 ⊢ (∀x∀y(φ → ψ) → {⟨x, y⟩ ∣ φ} ⊆ {⟨x, y⟩ ∣ ψ})
14	12, 13	impbii 180	1 ⊢ ({⟨x, y⟩ ∣ φ} ⊆ {⟨x, y⟩ ∣ ψ} ↔ ∀x∀y(φ → ψ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   ∈ wcel 1710   ⊆ wss 3258  ⟨cop 4562  {copab 4623

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-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624

This theorem is referenced by: (None)