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Theorem cbvmovw 2602
Description: Change bound variable. Uses only Tarski's FOL axiom schemes. See cbvmo 2605 and cbvmow 2603 for versions with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 9-Mar-1995.) (Revised by Gino Giotto, 30-Sep-2024.)
Hypothesis
Ref Expression
cbvmovw.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvmovw (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvmovw
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbvmovw.1 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
2 equequ1 2028 . . . . 5 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
31, 2imbi12d 345 . . . 4 (𝑥 = 𝑦 → ((𝜑𝑥 = 𝑧) ↔ (𝜓𝑦 = 𝑧)))
43cbvalvw 2039 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑦(𝜓𝑦 = 𝑧))
54exbii 1850 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) ↔ ∃𝑧𝑦(𝜓𝑦 = 𝑧))
6 df-mo 2540 . 2 (∃*𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
7 df-mo 2540 . 2 (∃*𝑦𝜓 ↔ ∃𝑧𝑦(𝜓𝑦 = 𝑧))
85, 6, 73bitr4i 303 1 (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1537  wex 1782  ∃*wmo 2538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-mo 2540
This theorem is referenced by:  cbveuvw  2606  cbvrmovw  3385
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