Theorem dropab2 45198

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 4844	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ⟨𝑧, 𝑥⟩ = ⟨𝑧, 𝑦⟩)
2	1	sps 2224	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → ⟨𝑧, 𝑥⟩ = ⟨𝑧, 𝑦⟩)
3	2	eqeq2d 2777	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 = ⟨𝑧, 𝑥⟩ ↔ 𝑤 = ⟨𝑧, 𝑦⟩))
4	3	anbi1d 643	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)))
5	4	drex1 2476	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)))
6	5	drex2 2477	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧∃𝑥(𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑) ↔ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)))
7	6	abbidv 2832	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑤 ∣ ∃𝑧∃𝑥(𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)})
8		df-opab 5179	. 2 ⊢ {⟨𝑧, 𝑥⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑧∃𝑥(𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑)}
9		df-opab 5179	. 2 ⊢ {⟨𝑧, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)}
10	7, 8, 9	3eqtr4g 2826	1 ⊢ (∀𝑥 𝑥 = 𝑦 → {⟨𝑧, 𝑥⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜑})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812  {cab 2744  ⟨cop 4600  {copab 5178

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-13 2407  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-opab 5179

This theorem is used by: (None)