Theorem ax7v 1923

Description: Weakened version of ax-7 1922, with a dv condition on 𝑥, 𝑦. This should be the only proof referencing ax-7 1922, and it should be referenced only by its two weakened versions ax7v1 1924 and ax7v2 1925, from which ax-7 1922 is then rederived as ax7 1930, which shows that either ax7v 1923 or the conjunction of ax7v1 1924 and ax7v2 1925 is sufficient.

In ax7v 1923, it is still allowed to substitute the same variable for 𝑥 and 𝑧, or the same variable for 𝑦 and 𝑧. Therefore, ax7v 1923 "bundles" (a term coined by Raph Levien) its "principal instance" (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) with 𝑥, 𝑦, 𝑧 distinct, and its "degenerate instances" (𝑥 = 𝑦 → (𝑥 = 𝑥 → 𝑦 = 𝑥)) and (𝑥 = 𝑦 → (𝑥 = 𝑦 → 𝑦 = 𝑦)) with 𝑥, 𝑦 distinct. These degenerate instances are for instance used in the proofs of equcomiv 1928 and equid 1926 respectively. (Contributed by BJ, 7-Dec-2020.) Use ax7 1930 instead. (New usage is discouraged.)

Assertion

Ref	Expression
ax7v	⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))

Distinct variable group:   𝑥,𝑦

Proof of Theorem ax7v

Step	Hyp	Ref	Expression
1		ax-7 1922	1 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-7 1922

This theorem is referenced by:  ax7v1  1924  ax7v2  1925