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Theorem cbvmow 2601
Description: Rule used to change bound variables, using implicit substitution. Version of cbvmo 2602 with a disjoint variable condition, which does not require ax-10 2139, ax-13 2375. (Contributed by NM, 9-Mar-1995.) (Revised by GG, 23-May-2024.)
Hypotheses
Ref Expression
cbvmow.1 𝑦𝜑
cbvmow.2 𝑥𝜓
cbvmow.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvmow (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvmow
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbvmow.1 . . . . 5 𝑦𝜑
2 nfv 1912 . . . . 5 𝑦 𝑥 = 𝑧
31, 2nfim 1894 . . . 4 𝑦(𝜑𝑥 = 𝑧)
4 cbvmow.2 . . . . 5 𝑥𝜓
5 nfv 1912 . . . . 5 𝑥 𝑦 = 𝑧
64, 5nfim 1894 . . . 4 𝑥(𝜓𝑦 = 𝑧)
7 cbvmow.3 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
8 equequ1 2022 . . . . 5 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
97, 8imbi12d 344 . . . 4 (𝑥 = 𝑦 → ((𝜑𝑥 = 𝑧) ↔ (𝜓𝑦 = 𝑧)))
103, 6, 9cbvalv1 2342 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑦(𝜓𝑦 = 𝑧))
1110exbii 1845 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) ↔ ∃𝑧𝑦(𝜓𝑦 = 𝑧))
12 df-mo 2538 . 2 (∃*𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
13 df-mo 2538 . 2 (∃*𝑦𝜓 ↔ ∃𝑧𝑦(𝜓𝑦 = 𝑧))
1411, 12, 133bitr4i 303 1 (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535  wex 1776  wnf 1780  ∃*wmo 2536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-11 2155  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  df-mo 2538
This theorem is referenced by:  cbveuw  2604  cbvrmow  3407  dffun6f  6581  opabiotafun  6989  2ndcdisj  23480  cbvdisjf  32591  phpreu  37591  mo0sn  48664  isthincd2lem1  48827
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