Theorem alcomw 2078

Description: Weak version of alcom 2197 and biconditional form of alcomimw 2076. Uses only Tarski's FOL axiom schemes. (Contributed by BTernaryTau, 28-Dec-2024.)

Hypotheses

Ref	Expression
alcomw.1	⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓))
alcomw.2	⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒))

Assertion

Ref	Expression
alcomw	⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑦∀𝑥𝜑)

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

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

Proof of Theorem alcomw

Step	Hyp	Ref	Expression
1		alcomw.2	. . 3 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒))
2	1	alcomimw 2076	. 2 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
3		alcomw.1	. . 3 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓))
4	3	alcomimw 2076	. 2 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
5	2, 4	impbii 212	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 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  unissb  4908  dftr2c  5223  cotrg  6113  cnvsym  6116  dffun2  6550  mh-unprimbi  37088  mh-infprim2bi  37091