Theorem rexraleqim 3328

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 2626	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑌 ↔ 𝑧 = 𝑌))
3	1, 2	imbi12d 334	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝜓 → 𝑥 = 𝑌) ↔ (𝜑 → 𝑧 = 𝑌)))
4	3	rspcva 3307	. . . . 5 ⊢ ((𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑌)) → (𝜑 → 𝑧 = 𝑌))
5		rexraleqim.2	. . . . . 6 ⊢ (𝑧 = 𝑌 → (𝜑 ↔ 𝜃))
6	5	biimpd 219	. . . . 5 ⊢ (𝑧 = 𝑌 → (𝜑 → 𝜃))
7	4, 6	syli 39	. . . 4 ⊢ ((𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑌)) → (𝜑 → 𝜃))
8	7	impancom 456	. . 3 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝜑) → (∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑌) → 𝜃))
9	8	rexlimiva 3028	. 2 ⊢ (∃𝑧 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑌) → 𝜃))
10	9	imp 445	1 ⊢ ((∃𝑧 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑌)) → 𝜃)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202

This theorem is referenced by:  cramerlem3  20495