Theorem rexraleqim 3577

Description: Statement following from existence and generalization with equality. (Contributed by AV, 9-Feb-2019.)

Hypotheses

Ref	Expression
rexraleqim.1	⊢ (𝑥 = 𝑧 → (𝜓 ↔ 𝜑))
rexraleqim.2	⊢ (𝑧 = 𝑌 → (𝜑 ↔ 𝜃))

Assertion

Ref	Expression
rexraleqim	⊢ ((∃𝑧 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 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

Step	Hyp	Ref	Expression
1		rexraleqim.1	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝜓 ↔ 𝜑))
2		eqeq1 2742	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑌 ↔ 𝑧 = 𝑌))
3	1, 2	imbi12d 345	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝜓 → 𝑥 = 𝑌) ↔ (𝜑 → 𝑧 = 𝑌)))
4	3	rspcva 3559	. . . . 5 ⊢ ((𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑌)) → (𝜑 → 𝑧 = 𝑌))
5		rexraleqim.2	. . . . . 6 ⊢ (𝑧 = 𝑌 → (𝜑 ↔ 𝜃))
6	5	biimpd 228	. . . . 5 ⊢ (𝑧 = 𝑌 → (𝜑 → 𝜃))
7	4, 6	syli 39	. . . 4 ⊢ ((𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑌)) → (𝜑 → 𝜃))
8	7	impancom 452	. . 3 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝜑) → (∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑌) → 𝜃))
9	8	rexlimiva 3210	. 2 ⊢ (∃𝑧 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑌) → 𝜃))
10	9	imp 407	1 ⊢ ((∃𝑧 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑌)) → 𝜃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070

This theorem is referenced by:  cramerlem3  21838