Theorem dropab1 40772

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 4796	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑦, 𝑧⟩)
2	1	sps 2180	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑦, 𝑧⟩)
3	2	eqeq2d 2832	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 = ⟨𝑥, 𝑧⟩ ↔ 𝑤 = ⟨𝑦, 𝑧⟩))
4	3	anbi1d 631	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
5	4	drex2 2460	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
6	5	drex1 2459	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑦∃𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
7	6	abbidv 2885	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑦∃𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)})
8		df-opab 5121	. 2 ⊢ {⟨𝑥, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑)}
9		df-opab 5121	. 2 ⊢ {⟨𝑦, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑦∃𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)}
10	7, 8, 9	3eqtr4g 2881	1 ⊢ (∀𝑥 𝑥 = 𝑦 → {⟨𝑥, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ 𝜑})

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1531   = wceq 1533  ∃wex 1776  {cab 2799  ⟨cop 4566  {copab 5120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-13 2386  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-opab 5121

This theorem is referenced by: (None)