Theorem dropab2 44417

Description: Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
dropab2	⊢ (∀𝑥 𝑥 = 𝑦 → {⟨𝑧, 𝑥⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜑})

Proof of Theorem dropab2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq2 4898	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ⟨𝑧, 𝑥⟩ = ⟨𝑧, 𝑦⟩)
2	1	sps 2186	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → ⟨𝑧, 𝑥⟩ = ⟨𝑧, 𝑦⟩)
3	2	eqeq2d 2751	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 = ⟨𝑧, 𝑥⟩ ↔ 𝑤 = ⟨𝑧, 𝑦⟩))
4	3	anbi1d 630	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)))
5	4	drex1 2449	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)))
6	5	drex2 2450	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧∃𝑥(𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑) ↔ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)))
7	6	abbidv 2811	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑤 ∣ ∃𝑧∃𝑥(𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)})
8		df-opab 5229	. 2 ⊢ {⟨𝑧, 𝑥⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑧∃𝑥(𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑)}
9		df-opab 5229	. 2 ⊢ {⟨𝑧, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)}
10	7, 8, 9	3eqtr4g 2805	1 ⊢ (∀𝑥 𝑥 = 𝑦 → {⟨𝑧, 𝑥⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜑})

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535   = wceq 1537  ∃wex 1777  {cab 2717  ⟨cop 4654  {copab 5228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-13 2380  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229

This theorem is referenced by: (None)