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Theorem rexraleqim 3579
 Description: Statement following from existence and generalization with equality. (Contributed by AV, 9-Feb-2019.)
Hypotheses
Ref Expression
rexraleqim.1 (𝑥 = 𝑧 → (𝜓𝜑))
rexraleqim.2 (𝑧 = 𝑌 → (𝜑𝜃))
Assertion
Ref Expression
rexraleqim ((∃𝑧𝐴 𝜑 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑌)) → 𝜃)
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝑌,𝑧   𝜑,𝑥   𝜓,𝑧   𝜃,𝑧
Allowed substitution hints:   𝜑(𝑧)   𝜓(𝑥)   𝜃(𝑥)

Proof of Theorem rexraleqim
StepHypRef Expression
1 rexraleqim.1 . . . . . . 7 (𝑥 = 𝑧 → (𝜓𝜑))
2 eqeq1 2799 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = 𝑌𝑧 = 𝑌))
31, 2imbi12d 346 . . . . . 6 (𝑥 = 𝑧 → ((𝜓𝑥 = 𝑌) ↔ (𝜑𝑧 = 𝑌)))
43rspcva 3557 . . . . 5 ((𝑧𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑌)) → (𝜑𝑧 = 𝑌))
5 rexraleqim.2 . . . . . 6 (𝑧 = 𝑌 → (𝜑𝜃))
65biimpd 230 . . . . 5 (𝑧 = 𝑌 → (𝜑𝜃))
74, 6syli 39 . . . 4 ((𝑧𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑌)) → (𝜑𝜃))
87impancom 452 . . 3 ((𝑧𝐴𝜑) → (∀𝑥𝐴 (𝜓𝑥 = 𝑌) → 𝜃))
98rexlimiva 3244 . 2 (∃𝑧𝐴 𝜑 → (∀𝑥𝐴 (𝜓𝑥 = 𝑌) → 𝜃))
109imp 407 1 ((∃𝑧𝐴 𝜑 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑌)) → 𝜃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1522   ∈ wcel 2081  ∀wral 3105  ∃wrex 3106 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-v 3439 This theorem is referenced by:  cramerlem3  20982
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