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Theorem alexeqg 3623
Description: Two ways to express substitution of 𝐴 for 𝑥 in 𝜑. This is the analogue for classes of sb56 2277. (Contributed by NM, 2-Mar-1995.) (Revised by BJ, 27-Apr-2019.)
Assertion
Ref Expression
alexeqg (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem alexeqg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2832 . . . . 5 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
21anbi1d 631 . . . 4 (𝑦 = 𝐴 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝐴𝜑)))
32exbidv 1922 . . 3 (𝑦 = 𝐴 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
41imbi1d 344 . . . 4 (𝑦 = 𝐴 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝐴𝜑)))
54albidv 1921 . . 3 (𝑦 = 𝐴 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
6 sb56 2277 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
73, 5, 6vtoclbg 3548 . 2 (𝐴𝑉 → (∃𝑥(𝑥 = 𝐴𝜑) ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
87bicomd 225 1 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1535   = wceq 1537  wex 1780  wcel 2114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070  df-cleq 2813  df-clel 2891
This theorem is referenced by:  ceqex  3624  sbc6g  3781
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