Theorem ax8v 2145

Description: Weakened version of ax-8 2144, with a disjoint variable condition on 𝑥, 𝑦. This should be the only proof referencing ax-8 2144, and it should be referenced only by its two weakened versions ax8v1 2146 and ax8v2 2147, from which ax-8 2144 is then rederived as ax8 2148, which shows that either ax8v 2145 or the conjunction of ax8v1 2146 and ax8v2 2147 is sufficient. (Contributed by BJ, 7-Dec-2020.) Use ax8 2148 instead. (New usage is discouraged.)

Assertion

Ref	Expression
ax8v	⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))

Distinct variable group:   𝑥,𝑦

Proof of Theorem ax8v

Step	Hyp	Ref	Expression
1		ax-8 2144	1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-8 2144

This theorem is used by:  ax8v1  2146  ax8v2  2147