Theorem ax9v 2158

Description: Weakened version of ax-9 2157, with a disjoint variable condition on 𝑥, 𝑦. This should be the only proof referencing ax-9 2157, and it should be referenced only by its two weakened versions ax9v1 2159 and ax9v2 2160, from which ax-9 2157 is then rederived as ax9 2161, which shows that either ax9v 2158 or the conjunction of ax9v1 2159 and ax9v2 2160 is sufficient. (Contributed by BJ, 7-Dec-2020.) Use ax9 2161 instead. (New usage is discouraged.)

Assertion

Ref	Expression
ax9v	⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))

Distinct variable group:   𝑥,𝑦

Proof of Theorem ax9v

Step	Hyp	Ref	Expression
1		ax-9 2157	1 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-9 2157

This theorem is used by:  ax9v1  2159  ax9v2  2160