Theorem dropab2 44684

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 4830	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ⟨𝑧, 𝑥⟩ = ⟨𝑧, 𝑦⟩)
2	1	sps 2192	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → ⟨𝑧, 𝑥⟩ = ⟨𝑧, 𝑦⟩)
3	2	eqeq2d 2747	. . . . . 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 2445	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)))
6	5	drex2 2446	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧∃𝑥(𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑) ↔ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)))
7	6	abbidv 2802	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑤 ∣ ∃𝑧∃𝑥(𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)})
8		df-opab 5161	. 2 ⊢ {⟨𝑧, 𝑥⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑧∃𝑥(𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑)}
9		df-opab 5161	. 2 ⊢ {⟨𝑧, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)}
10	7, 8, 9	3eqtr4g 2796	1 ⊢ (∀𝑥 𝑥 = 𝑦 → {⟨𝑧, 𝑥⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜑})

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780  {cab 2714  ⟨cop 4586  {copab 5160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-13 2376  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-opab 5161

This theorem is referenced by: (None)