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Theorem rexraleqim 3591
 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 2805 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = 𝑌𝑧 = 𝑌))
31, 2imbi12d 348 . . . . . 6 (𝑥 = 𝑧 → ((𝜓𝑥 = 𝑌) ↔ (𝜑𝑧 = 𝑌)))
43rspcva 3572 . . . . 5 ((𝑧𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑌)) → (𝜑𝑧 = 𝑌))
5 rexraleqim.2 . . . . . 6 (𝑧 = 𝑌 → (𝜑𝜃))
65biimpd 232 . . . . 5 (𝑧 = 𝑌 → (𝜑𝜃))
74, 6syli 39 . . . 4 ((𝑧𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑌)) → (𝜑𝜃))
87impancom 455 . . 3 ((𝑧𝐴𝜑) → (∀𝑥𝐴 (𝜓𝑥 = 𝑌) → 𝜃))
98rexlimiva 3243 . 2 (∃𝑧𝐴 𝜑 → (∀𝑥𝐴 (𝜓𝑥 = 𝑌) → 𝜃))
109imp 410 1 ((∃𝑧𝐴 𝜑 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑌)) → 𝜃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2112  ∀wral 3109  ∃wrex 3110 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2794  df-clel 2873  df-ral 3114  df-rex 3115 This theorem is referenced by:  cramerlem3  21298
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