Theorem ax8dfeq 33038

Description: A version of ax-8 2112 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)

Assertion

Ref	Expression
ax8dfeq	⊢ ∃𝑧((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑦))

Proof of Theorem ax8dfeq

Step	Hyp	Ref	Expression
1		ax6e 2397	. 2 ⊢ ∃𝑧 𝑧 = 𝑤
2		ax8 2116	. . . 4 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑥 → 𝑧 ∈ 𝑥))
3	2	equcoms 2023	. . 3 ⊢ (𝑧 = 𝑤 → (𝑤 ∈ 𝑥 → 𝑧 ∈ 𝑥))
4		ax8 2116	. . 3 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝑦 → 𝑤 ∈ 𝑦))
5	3, 4	imim12d 81	. 2 ⊢ (𝑧 = 𝑤 → ((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑦)))
6	1, 5	eximii 1833	1 ⊢ ∃𝑧((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑦))

Syntax hints:   → wi 4  ∃wex 1776

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-12 2173  ax-13 2386

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777

This theorem is referenced by: (None)