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Theorem rexraleqim 3529
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 2817 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = 𝑌𝑧 = 𝑌))
31, 2imbi12d 335 . . . . . 6 (𝑥 = 𝑧 → ((𝜓𝑥 = 𝑌) ↔ (𝜑𝑧 = 𝑌)))
43rspcva 3507 . . . . 5 ((𝑧𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑌)) → (𝜑𝑧 = 𝑌))
5 rexraleqim.2 . . . . . 6 (𝑧 = 𝑌 → (𝜑𝜃))
65biimpd 220 . . . . 5 (𝑧 = 𝑌 → (𝜑𝜃))
74, 6syli 39 . . . 4 ((𝑧𝐴 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑌)) → (𝜑𝜃))
87impancom 441 . . 3 ((𝑧𝐴𝜑) → (∀𝑥𝐴 (𝜓𝑥 = 𝑌) → 𝜃))
98rexlimiva 3223 . 2 (∃𝑧𝐴 𝜑 → (∀𝑥𝐴 (𝜓𝑥 = 𝑌) → 𝜃))
109imp 395 1 ((∃𝑧𝐴 𝜑 ∧ ∀𝑥𝐴 (𝜓𝑥 = 𝑌)) → 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2157  wral 3103  wrex 3104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-rex 3109  df-v 3400
This theorem is referenced by:  cramerlem3  20712
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