Theorem dropab2 44467

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 4874	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ⟨𝑧, 𝑥⟩ = ⟨𝑧, 𝑦⟩)
2	1	sps 2185	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → ⟨𝑧, 𝑥⟩ = ⟨𝑧, 𝑦⟩)
3	2	eqeq2d 2748	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 = ⟨𝑧, 𝑥⟩ ↔ 𝑤 = ⟨𝑧, 𝑦⟩))
4	3	anbi1d 631	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)))
5	4	drex1 2446	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)))
6	5	drex2 2447	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧∃𝑥(𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑) ↔ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)))
7	6	abbidv 2808	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑤 ∣ ∃𝑧∃𝑥(𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)})
8		df-opab 5206	. 2 ⊢ {⟨𝑧, 𝑥⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑧∃𝑥(𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑)}
9		df-opab 5206	. 2 ⊢ {⟨𝑧, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)}
10	7, 8, 9	3eqtr4g 2802	1 ⊢ (∀𝑥 𝑥 = 𝑦 → {⟨𝑧, 𝑥⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜑})

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779  {cab 2714  ⟨cop 4632  {copab 5205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2377  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-opab 5206

This theorem is referenced by: (None)