Theorem ax11w 2165

Description: Weak version of ax-11 2192 from which we can prove any ax-11 2192 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-11 2192, this theorem requires that 𝑥 and 𝑦 be distinct i.e. are not bundled. It is an alias of alcomimw 2073 introduced for labeling consistency. (Contributed by NM, 10-Apr-2017.) Use alcomimw 2073 instead. (New usage is discouraged.)

Hypothesis

Ref	Expression
ax11w.1	⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ax11w	⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

Distinct variable groups:   𝑦,𝑧   𝑥,𝑦   𝜑,𝑧   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜓(𝑥, 𝑧)

Proof of Theorem ax11w

Step	Hyp	Ref	Expression
1		ax11w.1	. 2 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))
2	1	alcomimw 2073	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1810

This theorem is used by: (None)