Theorem bnj1465 35542

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1465.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
bnj1465.2	⊢ (𝜓 → ∀𝑥𝜓)
bnj1465.3	⊢ (𝜒 → 𝜓)

Assertion

Ref	Expression
bnj1465	⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → ∃𝑥𝜑)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem bnj1465

Step	Hyp	Ref	Expression
1		bnj1465.3	. . . 4 ⊢ (𝜒 → 𝜓)
2	1	adantr 486	. . 3 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → 𝜓)
3		bnj1465.2	. . . . 5 ⊢ (𝜓 → ∀𝑥𝜓)
4		bnj1465.1	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	3, 4	bnj1464 35541	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
6	5	adantl 487	. . 3 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
7	2, 6	mpbird 260	. 2 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → [𝐴 / 𝑥]𝜑)
8	7	spesbcd 3834	1 ⊢ ((𝜒 ∧ 𝐴 ∈ 𝑉) → ∃𝑥𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  [wsbc 3742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-v 3456  df-sbc 3743

This theorem is used by:  bnj1463  35752