Theorem axextdfeq 35621

Description: A version of ax-ext 2697 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)

Assertion

Ref	Expression
axextdfeq	⊢ ∃𝑧((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ((𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)))

Proof of Theorem axextdfeq

Step	Hyp	Ref	Expression
1		axextnd 10634	. . 3 ⊢ ∃𝑧((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
2		ax8 2105	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤))
3	2	imim2i 16	. . 3 ⊢ (((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) → ((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)))
4	1, 3	eximii 1832	. 2 ⊢ ∃𝑧((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤))
5		biimpexp 35539	. . 3 ⊢ (((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ↔ ((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ((𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤))))
6	5	exbii 1843	. 2 ⊢ (∃𝑧((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ↔ ∃𝑧((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ((𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤))))
7	4, 6	mpbi 229	1 ⊢ ∃𝑧((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ((𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)))

Syntax hints:   → wi 4   ↔ wb 205  ∃wex 1774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-13 2366  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-clel 2803  df-nfc 2878

This theorem is referenced by: (None)