Theorem excomimw 2067

Description: Weak version of excomim 2200. Uses only Tarski's FOL axiom schemes. (Contributed by BTernaryTau, 23-Jun-2025.)

Hypothesis

Ref	Expression
excomimw.1	⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
excomimw	⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)

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

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

Proof of Theorem excomimw

Step	Hyp	Ref	Expression
1		excomimw.1	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))
2	1	notbid 321	. . . 4 ⊢ (𝑥 = 𝑧 → (¬ 𝜑 ↔ ¬ 𝜓))
3	2	alcomimw 2066	. . 3 ⊢ (∀𝑦∀𝑥 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑)
4	3	con3i 155	. 2 ⊢ (¬ ∀𝑥∀𝑦 ¬ 𝜑 → ¬ ∀𝑦∀𝑥 ¬ 𝜑)
5		2exnaln 1852	. 2 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑)
6		2exnaln 1852	. 2 ⊢ (∃𝑦∃𝑥𝜑 ↔ ¬ ∀𝑦∀𝑥 ¬ 𝜑)
7	4, 5, 6	3imtr4i 295	1 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1561  ∃wex 1802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803

This theorem is referenced by:  excomw  2069  dmcosseq  5959  dmcosseqOLD  5960  dfac5lem4  10098  cflem  10216