Theorem dropab1 44905

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
dropab1	⊢ (∀𝑥 𝑥 = 𝑦 → {⟨𝑥, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ 𝜑})

Proof of Theorem dropab1

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 4807	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑦, 𝑧⟩)
2	1	sps 2199	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑦, 𝑧⟩)
3	2	eqeq2d 2752	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 = ⟨𝑥, 𝑧⟩ ↔ 𝑤 = ⟨𝑦, 𝑧⟩))
4	3	anbi1d 638	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
5	4	drex2 2452	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
6	5	drex1 2451	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑦∃𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
7	6	abbidv 2807	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑦∃𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)})
8		df-opab 5138	. 2 ⊢ {⟨𝑥, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑)}
9		df-opab 5138	. 2 ⊢ {⟨𝑦, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑦∃𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)}
10	7, 8, 9	3eqtr4g 2801	1 ⊢ (∀𝑥 𝑥 = 𝑦 → {⟨𝑥, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ 𝜑})

Syntax hints:   → wi 4   ∧ wa 397  ∀wal 1546   = wceq 1548  ∃wex 1787  {cab 2719  ⟨cop 4564  {copab 5137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-13 2382  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-opab 5138

This theorem is referenced by: (None)