Theorem axextdfeq 36008

Description: A version of ax-ext 2709 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 10514	. . 3 ⊢ ∃𝑧((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
2		ax8 2120	. . . 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 1839	. 2 ⊢ ∃𝑧((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤))
5		biimpexp 35930	. . 3 ⊢ (((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)) ↔ ((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ((𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤))))
6	5	exbii 1850	. 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 230	1 ⊢ ∃𝑧((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ((𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)))

Syntax hints:   → wi 4   ↔ wb 206  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-13 2377  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-clel 2812  df-nfc 2886

This theorem is referenced by: (None)