Theorem ssoprab2i 5580

Description: Inference of operation class abstraction subclass from implication. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by set.mm contributors, 11-Nov-1995.) (Revised by set.mm contributors, 24-Jul-2012.)

Hypothesis

Ref	Expression
ssoprab2i.1	⊢ (φ → ψ)

Assertion

Ref	Expression
ssoprab2i	⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} ⊆ {⟨⟨x, y⟩, z⟩ ∣ ψ}

Distinct variable groups:   x,z   y,z

Allowed substitution hints:   φ(x,y,z)   ψ(x,y,z)

Proof of Theorem ssoprab2i

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssoprab2i.1	. . . . 5 ⊢ (φ → ψ)
2	1	anim2i 552	. . . 4 ⊢ ((w = ⟨x, y⟩ ∧ φ) → (w = ⟨x, y⟩ ∧ ψ))
3	2	2eximi 1577	. . 3 ⊢ (∃x∃y(w = ⟨x, y⟩ ∧ φ) → ∃x∃y(w = ⟨x, y⟩ ∧ ψ))
4	3	ssopab2i 4714	. 2 ⊢ {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)} ⊆ {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ ψ)}
5		dfoprab2 5558	. 2 ⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)}
6		dfoprab2 5558	. 2 ⊢ {⟨⟨x, y⟩, z⟩ ∣ ψ} = {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ ψ)}
7	4, 5, 6	3sstr4i 3310	1 ⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} ⊆ {⟨⟨x, y⟩, z⟩ ∣ ψ}

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ⊆ wss 3257  ⟨cop 4561  {copab 4622  {coprab 5527

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  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623  df-oprab 5528

This theorem is referenced by: (None)